Is it possible to ask Git to use CRLF instead of just LF at the end of the lines it puts into a file when it needs merging?
There is no setting which controls the line endings used for the "<<<<" markers in git; they are hardcoded to use '\n' in the git source code (see line 173 of xmerge.c).
If you set the "eol" or "core.eol" settings to "crlf", then the "<<<<" markers will have \r\n line endings in the file (this happens during the smudge/clean filter step, after the code linked above), but this has a major side-effect: the files will be "normalised" on their way into the repository, so you will commit files with unix line endings.
This is likely to be not what you want on a .Net project like the example above.
So I don't have a good answer for you, sorry.