How many is "a lot"? Is it plausible to rent a machine to do them all in a couple days, then buy a BX without a backhoe? Do you have other tasks for the backhoe?
I haven't had a BX backhoe, but I have had a small mini-ex, which most people will say is more capable than the BX. It can dig stumps, but it's very slow going. I'd do it for a couple, but I'd get pretty tired of it if there were a lot. A bigger machine will do them dramatically faster - instead of carefully working around the stump breaking the roots one by one, then slowly rolling it out, you can just put the bucket on the stump and pull.
If you're looking for a weekend hobby for a few years, then yes. If you just need the job done, I'd seriously consider renting a bigger machine (or hiring a guy with a machine who does it all for you in one go).