You will need at least 70mm if you want a full shot (head to toe) from 35' away. If you want a medium shot (waist up) then you will need closer to 150mm. If you're just wanting a fairly wide shot of the stage, then the Olympus 12-40mm f/2.8 will work great.
Correct, the zoom control feature on the ATEM Mini Pro only works with 4 servo lenses on the BMPCC 4K (and only 3 of them are still available new). I'm pretty sure they're all within your budget, but that also means they're fairly inexpensive, not the best quality, and have variable apertures that will make the image darker as you zoom in (or you'll have to close the aperture to the proper aperture to keep it from doing that, which will make everything darker throughout the zoom). Ultimately, these are not lenses I recommend, unfortunately.
Here is a list of those 4 lenses that will work on the BMPCC 4K and give you control of zoom through the ATEM Mini Pro: https://www.blackmagicdesign.com/support/faq/59009