Typically the bottom layer is rough plywood and the top layer is a nice finish grade.
If you have concerns about the overhand sagging (it probably won’t) you can place a trimmed down 2 x 6 UNDER the birch layer under the overhang.
Not sure what type of arm rail you are considering, but the standard 1×4 arm rest really added stability to the overhang.
As far as attaching the bar top, I suggest PL-200 or similar construction glue and a few finish nails, counter sunk and wood puttied, just like you thought.
The glue is very strong, just be sure you add some weights to the top while it sets up.
I would personally screw down the rough base layer, then just glue the finish layer to the rough base layer…got that?
Any way will work and it’s really your choice.
Cheers!
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