Yeah, if you were there during the initial Furfrou event, you might have been able to pick up everything besides Heart trim via a combination of Go and the GTS, but they haven't really been available since the event ended and especially not now that the season has changed over. Prolonging the Furfrou drought would probably be one of the biggest practical effects of Sword and Shield only getting the BDSP/Legends Pokémon instead of all 900+ (aside from the whole, "there are Pokémon you can put into Home that you can't take out" situation), which it will get frustratingly close to regardless. Especially because any reprieve provided by Niantic will probably be temporary at best (though, on that subject, the exclusion of Heart Trim specifically makes me think it's probably going to come back for a bit around Valentine's Day).
Fortunately, the only other Pokémon impacted by this mechanics change is Shaymin, which is both Mythical and already in BDSP, so it's not going to impact the GTS and will be fully available once the BDSP/Legends Home update rolls out.