Hmmm, both pickups open? There are two tests in that video, A and B. In A the Jazz has both pickups on, test B it's only using the neck pickup.
The difference with the Jazz having both pickups open (test A) is pretty sizeable but with just the neck pickup (test B) it's close enough to a P-bass sound, especially with a little bit of EQ tweaking.
That said... I'd suggest the OP try either the Jazz with the neck pickup solo'd or the SR800 with the bass set flat and the treble cut at least half way. (Might also want to experiment with having the volume knob backed off a bit.)