This is my dream console...but my house is full of radios and phonos that I think that ship sailed. I don't really hear of people fusing antique stereos, and I've never had any issues after replacing the caps.
This is no normal set - only doctors and the wealthy could have afforded these in the late 50s. The most basic radio had 5 tubes, ~9X% of record/radio combos would have 6-11 depending on quality. Concert grands had 40+ (unparalleled) and could heat a room lol.
Current inrush will be highest immediately after turning on the set when the caps have no stored charge. If not sized correctly I imagine the fuse might trip during warm up. If youre going to the trouble to fuse it you might want to look into other things too such as a voltage regulator depending on your power supply. Most sets where designed for 115-117 VAC and some utilities provide closer to 125-130 VAC these days resulting in sets running "hot" which can stress a components depending on factors. That being said, i've not had any issues after re-capping - the electrolytes being the most critical. in the 30 and 40s some sets had "death" caps. If this set has those (my death cap experience has been with 1940s philcos) i would replace with special caps that fail open rather than shorting, otherwise you probably dont have much to worry about.
I would ask this question at Antique Radio Forum as a misstep could cause damage to an irreplaceable set. Lots of antique radio and record player knowledge there.