Best debugging tool for my situation?

There is SPI block in QSYS, I think it is designed to be used with Nios II, but with some effort one can use it alone too. Perhaps it is easier to write SPI block by yourself in this case.

I am pretty sure, that you have huge timing problems in your design. Look here in the forum about TimeQuest timing analyzer, do the timing analysis, constrain the paths, learn maximal speed of your design and then perhaps you would need no SignalTap II anymore.