Hi Kevin,
The laser cutter will manage 6mm birch or poplar, so 4mm popler is not a problem. Yes the birch is a lighter wood and needs less energy to cut, but it is all around the same ballpark and the settings can be dialed in quickly enough. Also, yes, Purch costs are high at the moment and still fluctuating a lot.
One word of warning is buying generic plywood is that you don't know what is in the layers inside the wood, and because they are never seen they may contain knots and filler that will not cut as cleanly as the wood. This is why ply often has multiple quality marks, one being for the front, the back, and the intermediate. If the front is "bb" it is liley all the layers are not great to cut. This has never been a problem for me with the poplar I have used, but it is a common problem with the birch. Laser ply is generally just normal ply, but the intermediate layers are of a far better quality.
If you are going popular then you are probably okay with any supplier. I will say that the finish of birch is far superior, but that is a personal opinion. If the poplar has the finish and strength you want then go with that.
All the best,
Alistair