Welcome to the forum!
Unfortunately, I'm unable to help very much with EPS. For Windows, we have to install and configure Ghostscript, to be able to open EPS files. Since I don't use EPS, I've never taken those steps. So I can't open your test file.
Have you tried it with a test file, and investigated as to whether anything has actually been lost? And also what's being lost?
(I don't know this for sure, and certainly I'm not expert. But I have the impression that is sort of a standard warning, which may or may not actually apply to any given file. I have that impression, because I've seen that warning myself, in other situations besides EPS. But I've never actually seen where anything is lost. I'm guessing it refers to to code behind the image. But the image always looks fine to me.) (Although of course I understand your need to be sure.)
Since SVG is Inkscape's native format, I would probably try saving as SVG, and then save as EPS. Maybe that would avoid the loss?