Feeders never have to be larger than the service conductors.
Not when applying exceptions allowed by 310.15(B)(6). This has been debated many times on this forum. If the feeder is not carrying the entire dwelling load then it may have to be larger than the service conductors supplying it.
As far as the 125 amp breaker in the panel that only has 100 amp overcurrent device I see no problem with it. The max overcurrent protection of any conductor or device is not exceeded. This is no different than putting a 100 amp main breaker panel in a detached garage because you need a main and it cost less to just use a main breaker panel but you are supplying it with only a 60 amp feeder.
If the feeder supplied by 125 amp breaker draws more than 100 amps the breaker protecting the panel is going to trip.
I would not put in the 125 amp breaker unless for some reason you already have one. It is likely to be more expensive than a 100 amp breaker.