replace the entire breaker with a GFCI breaker
That is certainly a functional solution. A double-pole GFCI breaker is not cheap - you can typically get several GFCI receptacles for the price of one GFCI breaker. Remember also that you only need to replace the first receptacle in each chain with a GFCI receptacle to get full protection, though you are required (cough cough) to label all the following receptacles to indicate that they are GFCI protected (good) and no ground (if you replace them with 3-prong receptacles in order to allow use of 3-prong plugs, which you can do in this situation).
The one exception where it makes senses with an MWBC. If you have an MWBC then, depending on the wiring scheme, you may find that extra work is needed to properly split up the circuit so that you can install GFCI receptacles. In that case, GFCI breaker really makes a lot of sense.
If you have a main breaker or a disconnect before the relevant panel in order to safely cut power to the panel you are working on then replacing the breaker should be very easy to do. Electricians routinely install or replace breakers with the panel live, but turning it off first is not an unreasonable thing to do and could help avoid problems, particularly if the panel box is crowded.
barely fitting the GFCI's. I've been covering the terminals with some electrical tape to make sure that they don't accidentally contact the steel boxes
Yes, that's perfectly fine. As long as they really do fit. I have some boxes in my 1950s home which couldn't even fit a modern regular receptacle, let alone a GFCI receptacle, and I replaced the boxes. Of course, replacing boxes is easy in an unfinished basement and not quite as easy in other rooms.
But wait, steel boxes? Before jumping to GFCI everywhere, check to make sure the boxes aren't grounded already. I thought I would need to use GFCI receptacles to add grounded receptacles in several locations in my 1950s house and, so far at least, every single one of them turned out to have a good ground. But in some cases that was via a ground wire connected to the outside back of the box instead of being brought inside. Probably violated code, and I fixed them up to match the normal configuration, but the important thing is that the ground wire was there and working properly. (I haven't had to put in a GFCI receptacle in lieu of ground yet, though I have had to do plenty of other things, like fix hot/neutral reverses and switched neutrals and other craziness.)
