Can I gift a bush to someone who’s island has no space?

No, afraid not. If you visit an island and all your bush buttons are grayed out, even the specialty ones, then you know they have absolutely no available space. You have no recourse but to try again another day.

However, if you can select a bush just cannot place it (the check mark is gray), it means there is space elsewhere on the island; you just need to move the bush there. Until the developers fix the automatic space finder, you can move the bush manually or exit that bush move to a more spacious area and try again.