Picard does not currently support writing tags to WAV files:
https://tickets.metabrainz.org/browse/PICARD-41
You may want to convert your files to a format like FLAC that is audio wise identical to WAV, but was born with native tagging support. Picard handles FLAC files easily. 
(And they take up less space than WAVs too!)
(You should still be able to move/rename WAV files with Picard though. If it’s not doing that and you’re asking it to, that sounds like a bug!)