I don’t use pickup machines much so I can’t come up with an exact workflow for it without the OT in front of me, but it seems like they might be able to get pretty close to what you’re after.
Part 1 has a PUM on track 1 and flex machines playing their matching record buffers on tracks 2-4. Once you’ve got something recorded on the T1 PUM, you switch to part 2 which has flex machines on tracks 1, 3 and 4 and a PUM on T2. The record buffer on T1 still holds the audio you recorded in part 1, but now it’s being played back by a flax machine while you overdub on it in the PUM on track 2. Parts 3 and 4 move the PUM to T3 and T4 respectively.
Switch to a new bank of patterns and you could even continue the process on tracks 5-8, since you’d have four new parts to work with and record buffers 1-4 would already have loops recorded in them. When you were done recording you could either move to a third bank to get more free parts so you could start playing back all 8 tracks from flex machines and messing with them as you saw fit.
I might be missing something that’s obvious to people who use pickup machines regularly, and there are definitely details to work out, but I think the general idea should work.
EDIT: If you have the budget and desk space, you could always get something like one of the previous generation Electro-Harmonix loopers (2880/22500/45000 - they all seem to be about the same price used), sync it to the OT, record your loops in it, and then sample them on the fly back into the OT. The EHX loopers sound fine and are really easy to use (the 2880 is control-per-function and works more or less like an old Portastudio but with looping, couldn’t get much simpler and I feel like they still have a place because sometimes that kind of simplicity is really inspiring). Plus once you start controlling the looper with the OT’s MIDI sequencer, or bouncing audio back and forth between OT and looper, a whole lot of new possibilities open up. Something to consider, even though it’s not really what you were looking for.