First of all, cast-in-place plates or other "embeds" are common. Your contractor doesn't want to go through the effort to do it right! Yes, they can move if not properly set, but done correctly they provide better anchorage than the alternates.
I don't know the thickness of your door frame (likely in the 12 to 16 gage range), but welding light gage material to more massive material (the rebar) is tricky. It can be done, but it is a compromise to the proper method. Further, assuming you are in the US, you are crossing over the bounds of several codes (American Welding Society codes and standards for welding rebar, welding structural steel, and welding light gage materials are different and require different qualifications).
Depending upon the type of stainless steel you have, it might or might not be "weldable" in the context you have described. Your options for welding are limited in the field, as you can likely only use Shielded Metal Arc Welding (SMAW) or Flux Core Arc Welding (FCAW) processes. Other processes depend entirely on gas shielding for weld deposition quality and the conditions in the field are typically not conducive to these processes, though they can be done (GMAW or GTAW processes). Secondly, the filler metal must be compatible with both the stainless steel alloy and the mild steel rebar. Again, it can be done but is a bit tricky.
"Grouting" the rebar in place is not the proper method if you want to weld to it. Cementitious grout materials will be greatly affected at the point of the welding. Mechanical anchorage would be better. One alternate you could consider is to use a stainless steel wedge-type anchor bolt, install it through the frame, cut the head off if you have clearance problems, and weld the frame to the stainless steel bolt.
The proper way was to do it right to begin, using embedded plates. Anything else is a compromise.
