While I don't have a supporting link, my understanding is like yours, i.e. that your baggage allowance is governed by the FAREBASIS.
So the cabin class booked on XXX-FRA doesn't matter and it's not your friend's fault that LH doesn't offer any F on this leg.
...say you're on the waitinglist for F and wonder why in the world it never opens up on an intra-europe flight.

So if it's really a simple F thru-fare XXX to YYY in F with just a stopover in FRA, I think the point is clear and the argument at checkin in XXX is a rather short one that the passenger is allowed to 40k on both legs.

I think to get a definite answer, the link provided by Totti will not completely clarify your issue, so you'd either need a quick confirmation from LH (easy) or try to get access to the IATA ticketing rules yourself (not too much fun).

The thing to do (or rather not to do) would be to buy the official "Ticketing Handbook" from IATA. The trouble is, that at USD117 a copy, it's probably more expensive alone than paying the surcharge, sorry... And I cannot even guarantee it will bring 100% clarity.
The good thing if you buy one is that we could all start asking you all sorts of questions and you'll be the most liked person in the forum quickly.

Let's hope LH answers your question quickly or that someone's got a better idea.