Because of cost, I'll probably have to laminate 2 pieces of wood together to form the chop, rather than using a single thick piece.
This can be stronger and more stable than one solid piece so it's often a good way to go anyway. If the piece or pieces you get are flat-sawn, alternate the grain as in the vice bottom-left of the image in this Answer.
My question is this: given that I want to put dog holes (3/4") in the chop, how thick should I make it? I was thinking 2" - is that enough?
Yes 2" would be more than sufficient. You can install dogs in vice linings much thinner than this but for the style of face vice you're making thick chops are usual.
Since you are going with hardwood you should give thought to lining the chop with leather to help prevent marring of workpieces in softer woods.
Leather linings substantially increase grip at more modest clamping pressures anyway, reducing wear and tear on the mechanism. Not a big deal here but an important consideration with cheaper modern all-metal vices that aren't made as robustly as their vintage counterparts.