I've rinsed the BIAB bag with several different methods, and they all help with efficiency. I can get upper 80s or even 90+% efficiency if I rinse or sparge the bag.
Personally I have never squeezed a bag, I just let it drain by gravity and that's it. I'm afraid of squeezing out tannins or starches. It isn't necessary to risk this. You can get better extraction in other ways.
What I did for many years was a sort of Papazian old-style method, where I'd actually put the grain bag into a colander, heat up some water on the side to 170 F, then very slowly pour the water over the top of the bag in the colander. It's very time consuming, and basically constitutes a sort of fly sparge but without the sparge arm. Kind of a pain, but it will get the efficiency way up even into the 90s.
I have also toyed with the dunk sparge method. This also works well and is almost as good as the colander method, I can get maybe upper 80s with this method.
And yes, I have also reserved the dark roasted grains for the end of the mash, and this definitely works. I would add mine in like the last 5-10 minutes of the mash. However this can also reduce color contribution so you might need to use a little more than the recipe specifies to get the right color and enough flavor. I need to experiment more with this to get it just right.
These are all great ideas and deserve more experimentation to find out what works best for YOU.