I decided to try split dvc exp run command into repro & exp-save. Here is another question about it - Saving repo snapshot.
But finally I’ve gotten the same problem with the changing git-lfs endings after dvc exp save. It’s interesting that count of changes LFS files is not the same - sometimes it may be 3 files, sometimes only 1 file, but I did’t get all tracked files or none of them in any attempt.