What happens if you attempt a manual between the auto-double and manual-double timing? If you don't get locked out of the manual, then can't you piano across the buttons and get more than one attempt?

Yep, and this is very cool! You get multiple chances at your manual! It's kind of like SF4's plinking, except if you miss the window and get one of your pianoed buttons, you won't get the highest strength, but rather, the strength of the button that hits the window first. Your inputs are not frozen or locked out because you pressed a button during the "no fly zone".
To clarify what the poster is talking about, during your linker (or opener), there is a large window where you can input an auto-double at the beginning and for most of the duration, and there is a very short window at the end where you can input a manual. In between these two windows is a short-ish window where no inputs are accepted; you won't get an auto-double or a manual (take a look at the gap between the bars in this video: https://vid.me/nTLs).
[-----------double---------] [man]
So, let's say that's a heavy linker, and you're trying to hit a heavy manual. If you press HP 1 frame before that manual window, nothing comes out. But what happens if you press HP, MP, and LP in rapid fashion by drumming your fingers across the buttons on an arcade stick (called "piano"ing; you try to press the buttons 1 frame apart)? Well, if the very first button you press (HP) overlaps with the manual window, you'll get a HP manual as expected. However, if you press HP 1 frame too early by accident, then MP will be the first button that hits the window and a MP manual will come out! And if you're 2 frames early, a LP manual will come out!
[-----------double---------] [HML]
[-----------double---------] H[ML-]
[-----------double---------] HM[L---]
* If you press a H button too early but you've pianoed M and L, you'll still get a manual to come out.
Since heavy manual after heavy linker is usually around a 2 frame link, doing this pianoing to get a "backup manual" is actually very smart, and turns it into, effectively, a 4 frame link, as long as you're okay with sometimes not getting HP as your manual. Note that, of course, if you're trying to get backup manuals after MP linker, you can only piano MP down to LP, and there's no way to get backup manuals after a light linker (trying to plink LP into LK will get you a throw). And, naturally, if you're really late with your pianoing, your (HP) manual will just be blocked. You have to be slightly early for this technique to work.
After 1 minute of training mode practice, I was getting 100% of my manuals after heavy linker (most of them were HP, but the ones that weren't were either MP or LP). Pretty cool!