The terminology here matters, as you are not on a "connecting" flight from FRA-PMI.

I do not know how the baggage limit will apply here in what is akin to an open-jaw ticket. But if there isn't award availability on that leg showing on the Alaska site in business on that flight, there isn't award availability in business on that flight. (I have been watching that same route for awhile and not seen any business availability on that segment across various days and timings.).

But for others, for future reference, if you were merely connecting in FRA on your way to PMI, the business class baggage allowance would apply for the entire journey regardless of the class on the FRA-PMI flight.

As for your situation, you should be able to buy extra baggage online in the week between your flights.