Lol this is a good idea to do with your friends, but it is not a good idea to do in any serious tournament. You have to realize that this is logically flawed from the start. Just because one participant in a sport/game/etc. is better and/or winning, doesn't mean you should give the other person a large advantage to even the two out. Your interpretation of the purpose of counterpicking is wrong. A lot of people don't have a good grasp of the purpose of counterpicking, though. Some people don't like counterpicks and think neutrals is where it's at. Others love stages like Poke Floats and Jungle Japes and love to give a strong advantage to the lesser player.
I, personally, love stages like Brinstar, Mute City, and Rainbow Cruise. I totally understand problems with these stages and understand why they could be banned in a tournament because they have certain "unfair" qualities. However, I also believe that part of what makes Melee a great game is the stage diversity. And when a stage doesn't have qualities that give certain characters incredible advantages (i.e. infinites, large stages to camp, stages with multiple accesses that promote camping, random, significant stage transitions, random things that damage you thus making the damage unavoidable), it should be legal. The purpose of counterpicking is not only to have fun on more than one stage, but to also give the other player's character a possible advantage because, let's face it, the characters are not equal.
It's more like a home-field advantage kind of thing, to compare this to sports. The football field in Green Bay where it is freezing and snowing during the winter is the same length as the field in Miami where it floods sometimes. They both are also the same size as the artificial-turf field in Minnesota. These three are also the same size as Seattle's field where the fans can create so much noise, the offense can barely hear the quarterback hiking the ball. Yet there are conditions that give advantages to teams when they play there, hence the term home-field advantage. Except for the case with Seattle (and excluding psychological advantages), these physical conditions become advantages to the players through experience in playing on the field routinely. (This is why a player may choose a stage that may not give his character an advantage, per se, but gives the player the upper hand by having lots of experience playing on a certain stage.
This is why a player is given a counterpick. This is also why a player does not get a counterpick that will completely change the game (eg. camping).
(I realize that someone may come in and say something stupid about baseball fields having size-variations in the outfields, but these variations are so small, that it does not change the basis of the gameplay the way camping does in Melee.)
When it comes to a stage like Rainbow Cruise, the stage moves, yes, but systematically and the same way every time. It is not random. In fact, if you study this stage well, you will know when things pop up. Careful play will prevent getting gimped. The only bad thing is that the sides are really short in distance so you can die really easily. People can be killed off the top at relatively low percentages too. However, I don't see too many character match ups that get greatly affected by this stage. (eg. With this stage legal, every Falcon player will always win because of X, Y, and Z.) It's not like this stage completely sways match ups. If anything it is nifty to use as a counterpick to show off your knowledge for this stage in terms of when platforms appear/disappear.
Mute City is a tough one to argue for, especially since I am a Jigglypuff main. Ledges is probably the biggest problem considering it gives a very clear advantage to Peach and Jigglypuff who never have to worry about recovery landing lag. Also, if you're only in reach to grab the ledge and there isn't one, then obviously you get screwed. I have heard people say that Melee is 40 percent edgeguarding (obviously a made-up statistic), but it still seems to hold truth. But this obviously doesn't apply when it comes to Armada vs. Mango. Melee at one point may have been a lot more focused on edgeguarding, but in the current high level, it is a lot more difficult for players to get grabs and simply gimp. I understand that having ledges helps in recovering options, but the metagame has moved, in a way, toward spacing, in my opinion. And so although the edgeguarding may get easier to kill(at only the latter half of the stage when there is no ground underneath), the overall basis of what the game is has not been compromised. The bulk of the game is played on the stage. Any Falco player will say they are screwed once they are off the ledge of a stage and I don't think it matters whether you're talking about FD or Mute City. Because what makes Falco a good character is the stuff that he can do on the stage. It's not like Jigglypuff is high tier solely on recovery ability. In fact, very little difference does that make. It only makes the difference in edgeguard situations which I already explained is little of the gameplay. Just think about how much time you could spend practicing your edgeguards on Falco when you play Marth but if you can't get a grab or get Falco off the ledge, then it doesn't matter. I feel like the cars are not a problem because you can see them coming and know at what point in the stage they occur.
Remember, you are given a ban for a reason. If you really don't like it, ban it. This is why I also think it's important to use the striking system for the neutral stages so that the first stage is as neutral as possible for both, seeing as how some neutral stages give clear advantages to characters. So if you get counterpicked in the second game and lose, then YOU get to counterpick and win the third game if you really are the better player that you may think you are. (Though, I do respect the opinions of those who say they have had personal experience where they BARELY lose in the first game, win the second by a large margin, but because they get counterpicked to Mute City they end up losing the match. Which I why I say ban it if you have that much trouble with it.) Again, I understand if I seem a little biased.
Brinstar's problem honestly is just the lava. There really isn't that much wrong with it. Except for arguably a weird platform set up, but I mean it's nothing like YS64 (which is a garbage stage btw). Falcons use this stage to counterpick so you can't really argue that it's a disadvantage purely for fast-fallers. The stage is about control, and so if the lava is coming up, jump up on the top platform and gain control. I can really only see people being upset with this stage because of Jiggs and I guess Peach. However, neither character has a meteor smash or spike to hit a character directly down into the lava when it is rising up above the bottom platform, thus resulting in a rest or some serious combos. The lava, theoretically, would only play a part when they are already off the stage. (Unless the fall into it on their own or just don't have enough time to get out of the way.) The point is that the majority of the time they get hit by the lava should be when off stage considering it's very visible.
One stage that I did not really like on the Counterpick list was Corneria. It does more than get rid of edgeguarding, it creates difficulty in killing off the sides because it's so big...which gives a clear advantage to Fox who has, needless to say, the best Upsmash in the game. I can't think of any other characters who would like to come to this stage for a counterpick. It's usually a fox (can't recall seeing a none-fox counterpicking, though I probably have seen a Falco or something, and yeah Pound 3 Mango to get his vendetta.) In any case, the bottom can be used for fox's infinite, the arwings will randomly shoot out of no where with no warning, and the level difference in platforms can cause problems with camping.
Not sure how I feel about Poke Floats since it is glitchy, has walls that you can bounce off of to stay alive, no ledges (not that you really need any in this stage though), walk-off areas, and can become a stage where you just camp really hard.
So, to return to Termina Bay, this stage has multiple accesses that promote camping. Because there are two levels, with two ways to get across, it makes it really easy to camp with projectiles, specifically Fox (or Falco). You can shoot one laser and then easily run away. If the other player picks Fox too, then it will just be a laser fest, completely changing the game, which is why I say this would be interesting for YOU to try out but not for tournament play. You may argue that people wouldn't stoop to that level to win, but if you are suggesting that this is done in the Grand Finals of a tournament, then yes people will definitely camp as hard as they can to win.
Final note, I mentioned camping a lot. I hope you understand that I mean excessive or infinite camping not camping as a legitimate play style or tactic that people choose to exercise on FD, PS, etc.
When I say camping, I am referring to stages that are big and weird and promote camping.
I'm on an amphetamine right now which is why this post turned into an essay.
TL;DR: This is a bad idea.
