how bad an idea is it to caulk around the base to stop the dirty water dribbling out onto the floor
If your toilet connects like this (as is common in the UK)
Caulking around the base (where it meets the floor) will have no effect.
In the UK, you can also almost certainly disregard advice about wax rings which are used in the US where plumbing arrangements differ.
In general if you have a leak, you need to find the source and replace the seal. In your case this might be the push-fit plastic flexible connector (7) between ceramic toilet (5,6) and plastic soil-pipe (8) below.
These seals cost around £6.
Obviously the issue is removing the close-coupled cistern and toilet. This is usually achievable on a DIY basis. Obviously there is something different about yours since your plumber mentioned concrete preventing them fixing the leak.
I suggest you edit the question above to insert photos showing 1) whole toilet 2) connection to soil pipe 3) how the toilet is attached to floor. That way people can offer better advice.
If your toilet is leaking somewhere underneath, you might be able to use caulk to temporarily contain a small pool of raw sewage within the outer perimeter of the toilet base but it isn't something I would be happy about, particularly if the floor is OSB or similar. Especially if this is upstairs.