How screwed am I?

I have never had a problem drilling out snapped bolts with left hand thread drill bits. Comes in a kit of six bits,start with the smallest,set drill for counter clockwise and SLOWLY drill into it. Use next size after hole is drilled and so on. Between the friction and the bits biting will come right out.
 
I bet you these T Slot bolts (or something similar) are used in that application. They are dropped into the retainer plate slot then the whole assembly tig welded to the tank.

Should be relatively easy to cut slots in the retainer plate to slide old bolt out and install replacements.
 
I'd take a dremel with a cut off wheel, and cut the little spot welds, each side of the broke bolts. Pry the metal piece up, insert new bolts, and clamp the metal piece back down, and re-tack weld. Fill the tank with water before welding, and it shouldn't go boom!
 
Easy fix. Grind out tack welds. remove plate. Take it to a respected shop and have it tacked back on with new bolts. Maybe a 5 minute job.
 
Inert gas Purging works a whole lot better. (argon) Any respected shop can perform this job. Should not cost much to fix either.
 
Quick video,I punch the pilot,and use the left hand bits starting from smallest and working up until it comes out, I don’t drill the pilot with a regular bit. I’ve got some pretty rusted and corroded bolts out this way. Sometimes will take a grinder to make it flat, rather than the punch and bit slide off. Good luck,total investment is about $20 cdn.
 
stick a hose in it venting the exhaust from some motor in it to displace the oxygen leave it in while welding