A couple of thoughts.
"Needless to say, no qualification of short circuit welds were produced." I will assume you are required to work within AWS D1.1 since that is the primary code that requires procedure qualification for Short Circuit GMAW.
"This particular shop produces heavy plate welds on a regular basis" If a shop regularly welds on heavy plate and cannot produce welding procedure specifications WPS's, Their quality program and management must be called into question before looking at the first weld.
You report that the visual profiles of the welds are different........... If they are different on top, and there are no procedures to direct the welders, my first assumption would be that they might be inconsistent at the root and sidewalls as well.
Distortion does not equal acceptable/compliant fusion.
Donated work and material... That's a shame. It's really too bad they did not send you in for a quick project review *before* production.
Fillet breaks or Cuts and Macros can verify individual workmanship, but I don't think that's what you are really after......
To start, I'd have to know the project specifications and requirements.
Regardless, with no Welding Procedures and no Welder Performance Qualification Reports, I'd tag it as Non-Conforming Material and document everything (reports stating lack of WPSs, pictures of all the welds, etc) and send it to the Engineer of Record and let them decide what they want to do going forward.
In my opinion, the very least is get the parameters they did use (Volts, Amps, Travel Speed, Wire designation, Shielding gas, any preheating, etc), have them run the Procedure Qualifying Tests. Based off those results, select a decent representative sample of the work completed and have it destructively tested (cut and Macroetch). Then submit all that to the EOR for acceptance.