Monosodium tartrate (sodium bitartate) is pretty easy to source, isn't it? I believe that would give sodium perchlorate and from that it's easy to get
the ammonium salt.
Or so I read, and it's been a long day; these aren't things I mess with.
Potassium bitartrate is even less soluble than potassium perchlorate. Perhaps a double displacement with ammonium bitartrate would work. Tartaric acid
is significantly easier to obtain than hexafluorosilicic acid.