Regrout is listed in Wiktionary and in the OED, though not, as far as I can tell, in any of the smaller Oxford dictionaries.
The word is not a new coinage: the earliest OED citation is from 1867, from the Annual Report of the Ohio Board of Public Works: "These and other [canal] locks have also been re-grouted and repaired in other respects." Most of the other examples also hyphenate the word, but the dictionary itself lists it as a single, unhyphenated, word.
I found an example of the word's use from a local (for me) firm, Melbourne Tile Restoration, who have a page about
Tile Regrouting on their website. They explain:
Regrouting is a task where the grout lines in between the tiles, floor and wall are removed and then new ones grouted.
So the word is real and ought to be allowed, along with
regrouted and
regrouting, although I doubt that the latter two will ever crop up in any puzzles.