A D20 would be more than enough, you just put the probabilities of the tiles in terms of 20 digits.

I'm saying if you wanted to mimic the happy/meh/sad faces on the die in this game you would need multiple dice. But since all of the percentages are in 5% increments yes a D20 is all you would need. I suggest a lookup table for the players who are uncomfortable or uninterested in doing the percentage division in their head, and the sum of the two lower partitions. But you've got me thinking you could probably also have a run of custom-labeled D20s with increments of 5 instead of 1 to eliminate the LUT.

The square would look like:

1-5 sad face 6-9 neutral face 10-20 happy face

more or less like an ability check in DnD

Yes, I think we're saying the same thing. Your second sentence is exactly what I mean by lookup table.

But you don't need a lookup table. The square itself would literally have that printed on it. No percentages.

but that is what a lookup table is -- something to turn the number on the die into the happy/meh/sad outcome. There would be nine on each card.