Tub drains I have seen do not need much vertical room directly below the actual hole in the bottom of the tub. Once through the floor, the drain usually then runs horizontally to meet the overflow pipe coming down from the end of the tub. You should be able to cut through the floor and notch the top of your joist just enough to install the drain without greatly compromising the strength of the joist ... and if you have only 2X6 or 2X8 joists, you can first add plates on the sides at the bottom.
My own clearance problem was at the point where the drain and overflow come together, and here is how I did (with sistered 2X6s) what I am suggesting you might be able to do:





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