I haven't tried upgrading only the slow HDD and leaving the fast SSD in the Surface Studio, but I imagine that you would need to do it as follows:
BEFORE replacing the HDD, install the latest version of IRST and use its SRT feature to "disable acceleration" on the stock rapid hybrid drive. Doing so will write all of the cached contents from the rapid hybrid drive's SSD onto its slow HDD. IMPORTANT NOTE: After disabling the acceleration DO NOT "reset to available" the SSD as doing so will destroy the 128/119 GB cache and you will NEVER be able to get it back!
Once you have disabled the acceleration, go ahead and replace the HDD with a faster/larger 2.5" SATA III SSD, and then run the Surface Studio recovery image (or restore the drive from backup, clone, etc.).
Once the Surface Studio is back up and running properly from the new 2.5" SATA III SSD, install the latest version of IRST and use its SRT feature to enable acceleration again. Doing that "should" get you back the rapid hybrid drive with its full 128/119 GB cache using the faster M.2 SSD and the slower SATA III SSD.
Again though, I've never actually tested this, so I don't know for certain that it will work properly (i.e. proceed at your own risk!).
Now for your particular situation, if I were you, I'd reinstall the stock 2.5" HDD in your Surface Studio and attempt to get the rapid hybrid drive back to working order again. Once you're there, I'd then follow the above steps to "properly" upgrade the HDD to a SDD. I realize that's a PITA seeing as removing and replacing the 2.5" HDD is the most difficult part (seeing as you have to remove the CPU/GPU heat sink), but it might be the only way for you to get your Surface Studio back to the way you want it to be (i.e. with the 128/119 GB rapid hybrid drive working with the stock M.2 SDD and your new SATA III SDD).
NOTE: If you, at any point, used IRST to "release" the M.2 SSD from the rapid hybrid drive, then you are just plain out of luck as there will be no way for you to get back the full 128/119 GB cache (only 64 GB).
I hope that helps you out some.