I would do, but unfortunately they're all wildly different current ratings; wouldn't prove anything.
It may well help, this house if I look at the total socket load, in fact total consumption it is not that high.
You can see coffee before going to bed, coffee and a flask 11 am, then another coffee at 1 pm and yet another 3 pm, and a shower at 4 pm, clearly if it trips while making coffee, or having a shower you will have a good idea what has caused it, same goes for washing machines, tumble driers etc. But in the main my socket circuits are not loaded over 4 kW as amps that is 17 amp, so a socket circuit will likely run on a 16 or 20 amp MCB without a problem.
In the main we use 2.5 mm cable, which is rated approx 20 amp, so as a test swapping immersion with sockets assuming the immersion also has a FCU will not be much of a problem, in fact likely you could do without the immersion heater for the time of the test.
To trip a 32 amp type B MCB with a short circuit it needs around 160 amp to flow, the 32 amp is for an extended time, and you can clearly work out if you have over 32 amp load on the circuit by adding the watts, ohms law watts = volt x amps x power factor, you can forget power factor it is nearly 1, so 7.36 kW to over load a 32 amp MCB, also of course a 13A fuse will go before a 32 amp MCB most although not all of the time. To trip a type B 32A MCB before blowing a 13 amp fuse, the current needs to be over 160 amp.
It seems most likely the MCB is faulty, so fitting a 16 or 20 amp is just to confirm what we think is likely cause.
If it were me, as I am not known for throwing money away, I would be looking for another make MCB which will fit. The problem is the buzz bar holes make to make are not standard, so fitting another make could mean the terminal clamp will not clamp well enough with a different make, so change the lot, then all the MCB's do match, clearly no longer a consumer unit as not type tested, now only a distribution unit, but that is one way to get around the problem.
Normally we have a good idea as to the state of the installation, rubber insulated cables point to needing a re-wire, or the plastersizer leaching out green gunge also points to needing a rewire, but wiring from mid 60's is normally still OK. I would use an insulation tester
to check for potential faults, one shown cost me £35 new, may be a bit more now.