I decided to have a try. What I needed to move is not the thin bracket that solders to the top and bottom tank, but the heavy L-bracket that connects those vertical parts to the core support. That took a fair bit more heat than my propane torch would have created (I used a small tip and a neutral-to-rich flame on my oxyacetylene setup).

So the brackets came off relatively easily, and I also found several square inches of rust where the previous "professional" hadn't cleaned or fully soldered them. That came off with a Scotchbrite disc, then I fluxed and retinned both sides. So far so good.

But I couldn't get a good bond when resoldering. The brackets tended to warp a bit with the heat (or maybe they were warped to start with), and due to the position I couldn't clamp them in place either...

Anyhow I now have them exactly where I want them. Now to find a radiator shop who can solder them on properly... and pressure-test to make sure my ham-handed efforts didn't create a new leak!