Yes. They only need his rights if they plan on using a Bird exception to re-sign him. If he got a minimum deal, they'd be using the minimum exception instead. Or they could use the room exception to give him a little more than that.
A couple of tweaks to your scenario can leave the Spurs with even more money to use to bring in players:
--Wait to re-sign Jack until the end of the off-season.
--Only give Splitter his qualifying offer. This is guaranteed to happen for at least the first couple of days of free agency. According to ShamSports (pretty much the best site there is for salary numbers), that hold would be $7.493 Million dollars. Using that instead of the $10 Million would give the Spurs a good deal of extra cap space.
--Re-sign Ginobili for about a million less per year. If you have to give him the third year with a partial guarantee, so be it.
--Finally, assume the cap is going to be about $65 Million. I don't know what you assumed before, but that's looking like a pretty accurate figure.
From looking at my really rough calculations, that should leave the Spurs with about $9.5 Million to spend on a free agent, while preserving the room exception and still being able to re-sign Jack. If the cap is at $62 Million, that still allows the Spurs to beat the MLE and to using the room exception.
Here's a link to Sham's site:
http://data.shamsports.com/content/p...ries/spurs.jsp