Machine shop
There wouldn't really be the need to splash out a load on a machine shop, Stainless sheet is available in various thicknesses, which could be cut to the correct size, carefully rolled around a dowel of the correct diameter, then inserted.
It may be a bit fiddly, but with patience would work well.
I've never had one quite as bad as this, but a few with "slack" headsets (can be removed by hand) If you shop arround, you can find some brands have much deeper insertion depths amd these usually alleviate the problem.