youbeyouibei FYI Baltic birch is birch veneers no poplar, just birch in and out. It is far stronger than popular/fir plys like oak that is just a veneer layer. All the standard 'hardwood' plywoods are structurally popular or fir cores. Okoume and Mahogony are wicked expensive they are the marine grades. I have done a lot of work with both super strong and tough, but for example the last order of Brynzeel AAA was $120 a sheet. It is quite expensive hard to justify as full marine grade is way over kill.
I have never had an issue with staining BB ply. Although I like plain birch personally so I have only stained it a few times typically I just varnish it. I would not use a single stain varnish but a varnish over stain. You want the outside layer to be plastic with no dyes.