Thank you, everyone, for your input!
I think it is quite clear that the transformer has at least one fault, even if only based on the fact the rest of the meter is functional while, with the transformer installed and powered through it, it trips the breaker.
At this stage, I see the following options:
1. Pay a company to rewind the transformer - I expect this to be costly (but happy to be contradicted).
2. Try to find the same transformer as spare (either NOS or from another broken 195a EDIT: or 196, by the looks of it /EDIT) - this would be ideal, but is likely to take a while, especially here in Europe where old Keithleys don't seem to be that common.
3. Look to find an equivalent replacement, either as one transformer or multiple, like bdunham7 mentions (I believe there should be enough room for two) - this is a more likely scenario and probably not as costly.
So, right now, I'm leaning towards option 3.
On with the search...