I would run ONE or the OTHER. I would not try to have both running off a tee; although I've never tried that let alone even thought of it. The reason for having a bypass valve (considering you have a stock ECU) is to allow air to be routed back into the intake system (before the turbo) where there is always a vacuum and not positive pressure (boost). If you run the Tial Q (vent to atmosphere) the ECU will not "see" the extra air coming from the BOV which in turn will make the engine run rich just at second you take you foot off the throttle coming off boost. As a result when shifting up in gears coming off boost in between those shifts, the engine/exhaust will backfire (or afterfire, which ever is the correct term) due to a rich mixture of fuel with no extra air to compensate that fuel; the fuel then ignites in the exhaust (hot) and fires.
If you run with the bypass valve, fine, okay, no problem; you can drive normal w/o stalling issues coming off boost.
If you run with the Tial Q BOV then be prepare to change driving habits when getting off boost. Ex: you see a red light 1/8 mile down the road and you're at 4000rpms in boost, don't press the clutch down from 4k rpms coming off boost and then brake; that will drop the rpms instantly and cause you to stall. Instead, get off the throttle (BOV will vent boosted air), brake (while still in gear) then release the clutch to neutral around 1500 +/- 500rpms.
Basically, if you have a vent to atmosphere BOV with a stock ECU, you need to get off boost while still in gear when coming to a stop or shift up quickly w/o letting the rpms drop too low/quickly; that will prevent the rpms from dropping instantly causing you to stall.
As for me, I downshift constantly, then release the clutch around 2k rpms (I have a Turboxs RFL BOV - vent to atmosphere).
Maybe I missed your point but I don't think there is any sense in running two different types of BOVs. Choose one that will suit you (sound), your Supra/engine (performance) and your driving habits.