Oil pressure will be highest with a stone cold engine. This is a relief valve, so popping out on start up shows you have oil pressure. Whether it moves in and out in use I don't know...ask the Triton Thrasher! I see Mosin is running a Triple, he'll know.
There are two types of Triumph valve. A fine gauze weak spring version used on all bikes except the triples, which have a stronger spring and a coarse gauze to aid identification. This will be the higher pressure type you have read about. Interchangeable so possible errors with inapproriate application.
Plenty of pictures available, try CBS website, US based old brit bike blog, featuring Triumph Service Bulletins. Google search Triumph Bonneville PRV Operation
Bit late now but the gap between plunger mudguard front and frame vertical tube is a good place for the filter. Toolbox fitment is too much hassle for me, sod originality, I go for simplicity. Easy access, oil runs a bit cooler, no contest.
Swarfy.