Have experience of this kind of situation / challenge.
Avoid the term 'risk assessment', although as short-cut jargon, I get it.
If you have lots of buildings, there will be variation between them, and whilst managing by regulation (asbestos, electricals etc) seems like a good idea, there will be massive differences between premises.
For example, a 10 storey block with 3 lifts, built in 2010 will have no asbestos, but there are lifts, other height issues (maybe roof cradles) and could well have aircon, basement etc. Another building could be pre-2000, (asbestos maybe) single storey, with fire shutters for a kitchen area. Etc.
A 'file' (almost a data sheet!) for each building can then form the basis for ongoing management (by FM, not you). Properly focussed on the particular hazards (features) in or around each premise.
Probably management by checklist works best for this. And of course fire risk assessment is by building, so no conflict there.
Important - don't forget the surrounding fences, walls, gates, and other structures (flagpoles, fountains, paths, lighting pylons). It's not about assessing, just logging and then managing the upkeep and repair of physical objects. Yes, building asset plans should be able to encompass this, but often don't do it thoroughly.
It's a recommendation I have made in the past, because 'by regulation' just wasn't working effectively. Cut-backs meant that 'condition surveys' were not done, so the structures themselves were not looked after. Engineers were 'let go' because the maintenance demands of lifts, hoists, cradles etc were unquantified. Discuss with your FM service as to how this is done now, maybe you're not far off this anyway.