Steve, this may be an opinion you don't what to have, ... but I think the welding shop is correct about not welding on the remaining portion of the frame. From looking at your photos I believe the frame broke in a brittle manner. A good quality steel should bend & stretch before finally failing, iron will fail in a brittle manner, steel shouldn't. Even in a localized brittle failure you should see evidence of stretching and pulling in some areas away from the initial brittle failure. I see little indication of bending in your photos, only a series of brittle failures, as confirmed on the flat portion of the cross member. That member should have torn and than bent before finally failing in tension, inside it broke without bending at all. The frame simply fractured (brittle like glass). The frame is obviously very thin and the slight bending action of the tow bar with time could make it even more brittle. Now, repair welding could easily make the remaining thin frame members even more brittle. In addition, the remaining frame member just looks too thin to receive a good quality weld. Think of it as putting super glue on tissue paper. Probably fish plates bolted to areas of the remaining frame members that could allow reassembly of the bumper and use the car would be fine for normal driving, but I wouldn't recommend pulling it again with the towbar.
Sorry, just my opinion.