How about if I do an IPC in an airplane on a day when weather is below
circling minimums at the aiport where the IPC is conducted?

That seems like an excellent way to assess an instrument pilot's
capabilities, yet there is no way to conduct a circling approach.

Should the conscientious CFII not do the IPC and thus miss this opportunity?

Should the conscientious CFII require a return to conduct a circling
approach on another day to complete the IPC?

I'd say this should be treated the same way one would treat a failure of the
glide slope transmitter - go to another airport where the requirement can be
met. If all requirements can't be met, don't sign off the IPC. I'm not
claiming this all makes sense, but it seems to be required.

And how do we reconcile this with the FAA Inspector's Handbook clearly
approves of an IPC in a Level 1 FTD which cannot perform circling
approaches?

I don't know. It sounds like whoever updated the PTS didn't talk to whoever's
in charge of the Handbook. Maybe there's some kind of waiver. The best way
to find out is probably to ask the FSDO that approved the FTD.