Read your AIP/Jepp
"Airways" was dropped from phraseology some time ago. You now only need to "REQUEST CLEARANCE"".
DAS, the only guidance you'll find is in ERSA for BK local procedures (unless there's some special arrangement between local operators and BK ATC). There doesn't appear to be any hard and fast rule of "when" to request clearance-only that you should call ready in the runup bay. That would probably be an appropriate time to obtain your clearance.
I suspect a scenario might go something like this (but stand to be corrected):
Regardless of whether you're VFR or IFR, if your flight is going to be subject to a clearance, you report taxi to SMC (with VFR details if not on a plan). I'd imagine (do they still have a COORD in the TWR?) this would set the wheels in motion for BK ATC to coordinate your clearance with SY, for when you "REQUEST CLEARANCE". They would most likely have it for you by the time you complete your taxi and runups (if needed).
There's not much in Jepp to provide guidance on this topic. As with most things, just use commonsense.
With regard to your students - guidance should be forthcoming from your CFI and the flying school's document suite. As long as you follow that consistently, you should be fine.