My suggestion would be to take it to a radiator shop and have them test it for pressure holding, flow and thermal efficiency. Servicing the radiator on a TR3 is a PITA once the front apron is installed, so it pays to do it right the first time. If it's basically in good shape, they can clean and paint it as well. My shop will do the pressure & flow tests for free, but you may have to pay for the thermal test.
I had a terrible overheating problem that was eventually traced to poor thermal efficiency; apparently the tubes were no longer in good thermal contact with the fins. But because of the unusual original design, where the tubes run through holes in the fins, the problem was not apparent on casual observation. A recore solved the problem permanently.
Last time I checked, a recore wasn't much cheaper than a new offshore reproduction. But I'm inclined to think the recore produces a better quality result, especially if they use a Modine core.