Who knows what might be possible if you designed a cell from scratch - perhaps you could rework all the machinery to access other parts of fold space. After all, there are some weird and wonderful machines out there like the 'Vault' (https://en.wikipedia.org/wiki/Vault_(organelle)) that can fit whole proteins inside them. Possibly a different cage-like structure could help fold designed proteins into as-before unseen structures.
Apparently I upvoted this question in the past (found it by searching for an answer - no AI, like the good old days)
https://biology.stackexchange.com/questions/2507/are-there-a...
One answer mentions actin and hexokinase. I'm not familiar with the actin fold, but looks like a bab sandwich of some kind.
Another commenter on this page mentioned the 'Rossman fold', another classic, and TIM barrels also occur to me. One caution is that some of these I would consider higher-level patterns - the 'Topology' level of CATH hierarchy.
Naturally, the more high-level (abstract) the fold pattern, the larger the sequence space it covers. It is less interesting to say that a helical bundle (for example) covers a lot of diverse sequences.