First: do NOT convert the FAT partition to NTFS because the Compaq diagnostics won’t work. This partition is on the system drive but is not part of drive C: – it is a separate partition with no drive letter.
No, you do not ‘have’ to have a FAT partition on the system drive but it makes it easier to run the Compaq diagnostics from their own partition.
The repair issue is not caused by there being a FAT partition on the system drive.
Try locating the two files in question and copying them to another name. Then rename the two files to something else and rename the copies to the original file names.
The set CHKDSK to run and reboot the system.
Let us know what happens.