That would be fine before the 2.0 update.
Ever since the 2.0 update, they've split up a single Pokemon file to individual components, in order to retain relevant information for SWSH, BDSP, and PLA.
So simply transferring from any other game may not be enough now; information pertaining to it first entering from GO would be missing.
I get a feeling HOME is viewing as default SWSH, and how SWSH handles BDSP-entered mons, is to wipe the met game off.
For your examples, let's say Monferno, mind hovering over it then taking a screenshot within HOME?
(I imagine it'll show the Galar symbol instead of the GO symbol)