The service manuals are on this site; I can't link for you, as I'm typing this via phone - but an easy way to locate them is to find any post by
@Mooseman - he has the link in his sig. Wiring diagrams included.
Now for the bad news...
If you cut any of the wires to the mirror, you will lose the power adjustment, the heater, or both (plus the turn signal, if you have them in the mirror housing.
It's not a separate wire(s) that control the memory feature, but a module that stores that, plus seat positon, etc., etc.
If you find different, let us know; I've been using my key for entry since I bought my Envoy over 2ys now. AFAIK, it's a bug for which there is no workaround except using the key to unlock with (locking w/ the fob is ok, for me at least.) Doesn't affect every owner, but it's not unheard of - you and I certainly aren't the only ones w/ the problem.
You might try using the power lock button instead of the fob. I do this because of the delay feature, and on the rare occasions where I do use the fob to unlock (as I did last night, loading via liftgate), about half the time, my mirrors don't move.
That suggests to me that there is *some* set of conditions that will trigger the bug (or not) - but no one has found it, that I know of. I know I haven't.
On edit: Actually, I did find something just now regarding this. It's on 'the other site' (aka: that which will not be named here).
It looks like there's a 'real' fix (which will be expensive, depending on how far you have to go), and a '10 cent' (permanent) workaround, which is to disable the seat / mirror repositioning in the DIC when the vehicle is unlocked. See:
here
I'll try the 'cheap' fix, which will work fine for me, as I'm the only one who drives my truck. If others drive yours - besides disabling the automatic sync, the new driver will likely need to push '1' or '2' on the door button, as well.