I'm sure a lot of this was already said, so my bad if I'm repeating anything. Just wanted to throw in my two cents.
First of all (@ OP), your suggested system of winner has to pick their character first followed by the loser picking character and stage allows for the loser to pick the absolute worst possible match-up combination for the winner (character and stage) if they wanted. While they likely won't pick a character they never use for a slightly better match up, this is still an unfair advantage at a competitive level and would just make sets likely come down to who wins game 1.
Unless you're a low tier main, the stage makes for the biggest match-up changes (probability wise, not gameplay), so it only makes sense that the stage should be the first one chosen for a CP. While the CP system is designed to give the losing player help, it's also not there to screw the winning player over. If a player can switch between characters and beat you on your stages, then he's better than you. It doesn't mean it needs to be changed more in your favor. There are basically two main ways that you can CP:
1. You CP a stage that is going to benefit the character you're going to use. This is your safest bet for a good counter-pick and is the one you should use if you don't have good secondaries (or also if you do). The fact that it's a good stage for you makes it less relevant what character your opponent picks, making it the safest choice.
2. You CP a stage with the purpose of hindering your opponent's character. This is a good option if your opponent is better than you or if you have a hard time with your opponent's character. Your opponent is more likely to switch characters this way than with method one, so you should be aware that this can just lead to a more neutral game, or in your case, actually to your opponents advantage.
Either way if your opponent is beating you regardless of how you CP, then as stated earlier he is better than you, or you just got outplayed. There are other fair ways I think you could set a counter-pick system up, but being able to freely choose the stage after characters are picked is not one of them.