The EICR is about safety, not complying with BS7671, it was at one time, but code 4 was removed as it was considered as confusing. In the main electric tripping does not produce a danger, so would not be flagged up.
Lighting has a maximum size of 16 amp for the trip, and in most houses since the ceiling roses are rated 6 amp, 6 amp is maximum size of a lighting circuit.
A RCD has a number of ratings, typical 10kA for how much is can safely switch off once when there is a fault, then 80 amp for how much it can handle on a continuous basis, and 30 mA the maximum imbalance it will allow without tripping and type AC, A, F, or B for how it can handle DC and high frequency AC.
The MCB also has three ratings the 10 kA the amps it will trip with in due time and the amps it will trip with within 0.01 seconds. So a B6 MCB will trip in time at 6 amp but very fast at 30 amp (5 x 6 amp) and a C6 very fast at 60 amp (10 x 6 amp).
So each circuit in theory you need to test to see what size the maximum MCB can be, so 30 amp at 230 volt needs an impedance of 230/30 = 7.66Ω pull some safety 5% so 7.28Ω and we use a loop impedance meter to check it.
The problem it the meters used are expensive, so not really worth the DIY guy knowing how to do it, as he is unlikely to have the test gear, and the MCB is not normally tested as such, so never use second hand as you don't know if they have been damaged.
With an incoming DNO fuse of 60 amp a 63A RCD would be OK, however we often have 80 or 100 amp supplies, so often we look as the total MCB load, not really cut and dried, but you could not say an inspector is wrong if he fails an installation because the MCB load is likely to exceed 63 amp, it is his judgement.
As the owner or landlord your job is simply to select some one who has the skill, you don't need to have the skill yourself, but I can see how the owner or landlord wants to know what is really required as there are so many trying to drum up trade and want to do unnecessary work, but many be the landlord should also worry if something goes wrong will he get the blame. The courts to me are biased towards the tenant when things go wrong. Even when it seems the tenant has not been very sensible. So I decided to sell old house not rent it out.
I can hardly believe some of the things tenants expect the county council to fix with council houses, and one does not want to be called because a RCD is tripping, so I would consider fitting RCBO's so no excuse for calling anyone out, but distance from house could change that, I had a 100 mile round trip.