Hi James,
This is a somewhat complicated issue, but I’ll do my best to explain. First thing you need to be aware of is that there are no off-peak paper or e-tickets within zones 1-6, apart from the off-peak day travelcard. The travelcard says that it is not valid on trains timed to depart after 0430 and before 0930, which means that it is definitely valid on the 0930 service. The freedom pass website says that it is valid after 0930. As it doesn’t specify by how much I would say that 0930 and a fraction of a second is fine, so to all intents and purposes it is valid on a 0930 service. I’d be even more confident given that trains rarely leave London Bridge early, especially at the end of the morning peak. The freedom pass doesn’t mention trains timed to depart, so if a late running train leaves after 0930 then you are fine.
The fares quoted on the National Rail website are contactless fares which are decided by touch in time rather than the time of the train. There is a small grace period of 3 minutes at the end of the morning peak, so in theory you would be able to catch a train at 0930, but London Bridge is quite a large station so they are probably erring on the side of caution by saying you’ll be charged a peak fare. I notice that the next train is at 0931 and they claim that will be off-peak. My worry is that people will touch in in good time and end up getting charged the peak fare anyway, but that’s not relevant to your enquiry.
I hope this helps.