DC Decouple the SA from the prescribed test point and assure the level of injected signal is well below the max of the SA input.? That will trace the BPF shape as the injected signal is swept through the BPF.? I don't have one, yet, but if it has a max hold, that would be appropriate as you sweep the injected signal.
The term "persistence" wrt a spectrum analyzer goes??? W A Y??? back to the "original" HP141 with its "variable persistence".? Yes, I used one many times.....