Turning a mic down accomplishes nothing as far as bleed is concerned. It will turn the WHOLE signal down, so everything stays relative.
I've definitely experimented with this. I had one of my teachers say something along the lines of when you think about a gain knob, substitute sensitivity. This may apply a lot more to live sound than recording but I've definitely played with similar results for recording drums.
We have all seen a mic pattern, whether it be a type of cardioid, omni, 8, what ever.
Pic if you haven't. Imagine when the gain setting is off, there is no pattern, because essentially you can't hear that signal. If you turned the gain up 3db you would then have a small pattern (no where near the size pictured), and then if you turned it up again, it would slightly larger, until you reached a gain level that would be picking up a lot more and would more sensitive to environment around it.
In live sound, this applies a lot to vocal mics. Where you have to worry about feed back and amplifying anything else on stage level and was a technique we were told if we were ever in a situation if we found ourselves without an EQ to combat feedback. So you set a very low gain level, so your "bubble" of a pattern is really small so you have to sing right up on the mic for it to be able to pick up any sound. Of course you can't leave it like this because the overall signal will be far to low in the mix with the rest of the instruments. So what you do is later down the chain, use a compressor or something with an output/makeup gain capability and simply increase the make up gain to a suitable level. This allows for the mic to stay at a level where it won't feedback and pick up anything else on stage, but the level is amplified down the chain to a level that will be able to be used.
Same sort of applies to recording drums. Your close micing your drums right, so you can apply the same concept. Have a low gain setting, so the mic is only picking up the majority of the snare sound. You can then later down the signal chain amplify the signal again to something usable, and low and behold, you'll have very very minimal bleed in that signal.
Just experiment with the idea, it can definitely help.