If you use cauls to clamp your veneered boards, any excess glue will try to ooze out a knot hole.
I like to have several layers of paper towel between the veneer and caul to absorb this kind of stuff so it gets drawn away from the veneer versus just spreading out on top of it.
For a large knot hole or really any hole larger than a pencil lead, the squeeze out can become a real issue.

When I make my marquetry type panels, there are all the seams to worry about for squeeze out. Since I make them on a sheet of vinyl self adhesive shelf paper, the glue can't get through and instead gets forced out the sides where it gets trimmed off with the saw.
Shelf paper is easy to peel off after pressing, but packing tape may work in a pinch.

When doing these marquetry panels (details in various blogs), I place the shelf paper sticky side up and attach the veneer to that. Then I roll on the glue to the substrate, join it all together and press between melamine cauls.
I have found that after laying it all out, I can fill areas or defects with a colored filler (resin/dyed glue/etc.) and then proceed as usual. The colored stuff stays in place and works well.

I'd never veneer edges since I see that fall off of stuff all the time. Especially a drawer where it will be scraped whenever the drawer is opened and closed.
No guarantees, but doing just one side may work out fine, but better with two.
I've used the TB2 dark for things before and it worked fine, just need to press the 'ell out if it to make sure it is all flat (it's kinda thick).
I just use an ink roller to spread out the glue, about like a goodly coat of latex paint on a wall. A fine notched trowel  works as well.