I'm not sure I have any 120mm frames. I think mine was 126mm for 5x2 gearing.
Anyway, I always remember quite a bit of space to the right of the freewheels, but perhaps that was also required for larger sprockets. So, it is possible that a narrow 6 speed freewheel would be close to fitting, especially if you go with 12T, or maybe 13T.
You may choose to cold-set your frame to 126mm, and be done with it.
Otherwise, I'd set your spacers with minimum spacing on the right, then adjust your left spacers, and see how it comes out.
I suppose you won't see the final dishing on the wheel until you're actually finished. Double check your derailleur hanger is straight.
You can also source a 5s freewheel, mount, and adjust the spacers and cut the axle.