I don't main MCH (secondary job), but for the first point I think it depends on skill speed / latency. Sometimes I see Slug pop just fine and go straight for it, sometimes I've had to hit Split again before realising the first one actually proc'd. If you weave a skill after the Split shot it's a bit easier to notice since is speeds up the application of your GCD move's effects (like how if you weave something after Clean Shot it won't wait until the flashy bullet delay animation to apply its damage). As for what's best to do if you don't have time to hit the proc in good time I would guess just go straight for another Split to not lose out on GCD time.

For the second point, I would try and avoid getting dry on TP in the first place. If I start getting very low I'll switch to single target while keeping Bishop out. If I am dry though, if Invigorate won't be available for a while then I'd make the switch to get some TP back. It also sort of depends on how much longer you'll need to keep aoe'ing for. Even with Promotion on - aoe skills won't be spammable with low tp, though they are much stronger than a pulse from Bishop turret.