Router Bit Recommendation

Forstner bit would be my choice, with these things in mind. They are best used in a drill press at slow speeds. If you are freehand drilling, it is hard to get the holes straight down, in which case, you can drill a hole in some ply at the drill press using that bit, then clamp the pre-drilled piece on your project and use it as a drill guide. You should use a variable speed drill at low speed. You want to hold it as steady as possible. Here is a sample:

You can use a regular drill bit, but it may not be as smooth a bore, and there could be tearout below, for this approach, consider using a brad point. The "wings" at the end will help make a good cut. But again, I'd make a drill guide to make sure the hole is vertical. Here's a sample:

Lots of ways to approach the job and all of the suggestions above would work with some limitations. Things to be considered include;
Are the ply faces very thin presentation veneers, or are the layers in the ply of equal thickness.
How are you going to establish and fix the hole positions?
How are you establishing the hole size, do you want a 3/4 bit and a direct plunge, or are you going to use a template and guide with a smaller bit, or a pattern bit and bearing, or are you content to drill with a forstener bit?

I believe that the simplest, least painful approach would be a decent 2 speed geared portable drill mounted in a drill stand and a 3/4 forstener bit, with a waste board underneath to drill into. This ticks pretty well all the boxes, the forstener should give a neat entry into the holes, and in combination with the backer board a neat exit on the other side, while the drill stand should ensure that the holes are perpendicular to the surface. You should also be able to align the point on the forstener to your positioning marks if you are simply measuring and marking positions.

You could use a 1/4 compression bit, guide bushing with a template, and a backer board. The opening in the template is larger than the guide bushing, you sit the router on the template, plunge through the ply and into the backer, then circulate the router around with the guide bushing making contact with the template to control the hole size. This would probably give you the best overall finish. Don't use a compression bit without a backer board, there is a possibility that the emerging tip will chip the back side unless it is supported. Also don't use a compression bit and try to work through in stages, the tip is an up spiral and will try and chip the face of the ply. If you plunge through the ply first close to the centre of the hole this can still happen, but as the fully engaged bit expands the hole to finished size, the chipped part is removed anyway.

For a direct plunge with a 3/4 bit, the bit needs to be a plunging bit, i.e. able to remove all of the diameter at once by having cutting faces on the entry end as well the sides. These are fairly rare, most bits in this size range are for profiling and only cut along the side, though some offer very shallow depth penetration as well. These bits can cut through the depth of a piece of material, but need to plunge down creating a ramp, not ideal for a small round hole.