Good question, here's my take:
From your post it appears the Welder is straight 240 volts, NO 120 VAC right???
I say that because you mentioned NOT running a Neutral (240 volt only, NOT 120) which is fine PROVIDED the Welder doesn't require 120 volts.
YOUR QUESTIONS:
"is it possible to blow the fuse on one hot leg and not the other?"
That's "possible" and if that happens, there's NO 240 volts to the welder. Sure, 120 volts exists between an unbroken Ungrounded (Hot) and the bare/green safety equipment GroundING Conductor, but if the welder doesn't use any 120 that shouldn't be an issue provided all is well and correct.
"I have a TIG that is pretty computerized and I'd hate to know what would happen if one leg of the 220 circuit dropped out."
If one leg drops out, there's no 240 to the welder, see the above. (See Notes below)
Now to using that 3 phase fused disconnect.
Having and using two of its three poles as a convenient disconnect will work and the fuses would be matched to the ampacity of the wires feeding it. The breaker in the panel is there to protect the feeders also, but additional fuses downstream (though not required) will not prevent the welder from working.
NOTES: 1) The two pole breaker will open BOTH conductors if there's an overload, and that will prevent the situation you mentioned of ONLY one leg being open to the welder.
2) If you want to possibly prevent or lessen the chances of only one fuse opening if theres an overload in a case where the breaker did NOT trip.............You might want to consider using dual element time delay (aka Slow Blow) fuses in the disconnect. HOWEVER all this depends on the time current curves of the fuses and breaker, as those change the chances of a fuse blowing but NOT the breaker or vice versa will change. Without getting into all the exact specific time current curves and calculations, I'm ONLY saying you may want to consider the use of a "Slow Blow" fuse IF YOURE WORRIED ABOUT ONLY ONE FUSE LEG OPENING. Again, if the welder is 240 ONLY and no 120 is used that shouldn't be a problem, but its just one possible way to insure the breaker opens which opens BOTH legs.
3) From the disconnect to the welder Id consider a section of flexible conduit.
SUMMARY if the welder is 240 ONLY (NO 120), no Neutral is required,,,,,,,,,,,,Its okay to use the disconnect although its not required to protect the feeders, the breaker already does that,,,,,,,,,,,,,If youre worried about opening only one leg (though shouldn't be a problem) consider slow blow fuses in the disconnect.
FINAL consult local authority and trained qualified professional electricians if in doubt and dont hang your hat or risk a hazard on my or any advice here lay or professional, NO WARRANTY ON ANYTHING I SAID I'm long retired and rusty on the latest codes.
Electrical and Legal Questions draw a lot of responses, some good, some perhaps possibly not so good, your choice and decision, so read and consider and do your homework and keep safe now.
John T Long retired EE