Hi 72Tiger,
The vast majority (if not all) spin-off oil filters are between the return pump and the oil tank (the frame in your case). There may be the rare oil filter on the feed side, but I know nothing about them. Others may.
Cannot comment on the cooler setup but I have my opinion on coolers for what it's worth. Everyone these days wants to cool the oil because the Jap bikes have them. Unlike the Jap bikes, the Triumphs bump along at 4 to 5k RPM, whereas those Asian bikes are uo around 11 to 13k RPM and more.
If you were cruising the highways at 5500RPM all day, you might justify a cooler, but the 650 Triumph simply will not stand this RPM for long periods. With modern multigrade oils and a filter, you'd have all the protection you'll ever need IMHO. No doubt the oil cooler pundits will want to maul me for my comments

. RR