Just had a little play with the donor cover and found that I can get 2 thou difference with the depth gauge by moderately pressing on the cover.
Success with the solder depends on whether the solder can create as much resistance as my pushing on the cover...
The two options I see are starting with the solder diametre as close as possible to the correct shim size, and/or use a softer solder.
Talk about splitting hairs.
Lord of the Bings