you can stick up some flux to make it tacky, line the bottom of the shield, then get a very thin solder strand and line the bottom, then press and heat, like making a sandwich. You can meter it very carefully this way.
I think they have solder down to 0.2 mm or whatever, so you can pick which one you think has the correct coverage.
tinning it to make absolutely sure there is no squeeze out might actually be harder because its hard to tell how much solder there is, but if you clean the flange with braid so its solder free, then line it with solder, you can control it better, maybe.
Or you MAYBE can get solder preform sheet, that is very thin, and cut a 'gasket' out of 4 pieces to line the bottom of the shield. Or run solder through a roller lol