Actually, I don't think either structure sounds very natural. It would sound better to say something like "That was a good idea that Kate had." But the distinction between your two examples is that if you say "Kate's good idea" it could imply that she's only ever had one good idea.

Actually, "That's Kate's good idea" would be fine in regular conversation and emphasizes that the idea came from Kate: "Was that your good idea?" "No, it was Kate's good idea." If you want to emphasize that the idea was a good one, rather than that it was Kate's, you would use the "good idea of Kate's" construction: "Was that one of Kate's ridiculous plans?" "No, it was a good idea of Kate's." I hope that helps!

You can say "That's Kate's good idea" and be grammatically correct, but it would require a very specific context for it to sound natural. I think it's because there are two separate ideas being expressed that usually wouldn't be combined in one without a reason. Without a reason to combine those ideas, it would probably be more natural to hear "That's a good idea. It was Kate's." or "That's Kate's idea. It's a good one." or something along those lines. Even if someone asked me "Whose good idea was that?" (which might sound sarcastic because of the odd combination of the two ideas) I would probably simply say "Kate's."

Bob (ironically): "Whose great idea was it again, for us to pitch our tent in a field with fresh cow pats in it?" Anne (looking pointedly at Kate, also ironically): "Kate's - that was Kate's great idea. Thank you Kate!" Why not "good idea" in this conversation? Maybe because "good" is not sufficiently emphatic to warrant such a laboured grammatical construction. Why not the present tense? Because Kate had and articulated the idea in the past, and the speakers are referring by implication to the specific moment when Kate had and/or articulated the idea. But in a specific context it could work: Bob: "Do you have any thoughts on how we get ourselves out of this pickle?" Tom: "Not that I've come up with myself, but Kate has come up with a couple of thoughts." Bob: "OK, like what then?" Tom: "Well, two ideas really, one good one and one not-so-good one. Kate's *good* idea is we park our cars between the cows and the tents while we pack our stuff away. Kate's *not-so-good* idea is for us to slap the cows on the rear until they stop trying to eat the stuff of ours that they've scattered everywhere." Here the present tense is used because it remains possible for the people in question to act on these ideas. "Good" can be used because it is sufficiently noteworthy to be mentioned in this way (and still feel natural) when contrasted (here explicitly, but could be implicitly) with "not-so-good" or "bad" or similar. So grammatically there is nothing per se wrong with "Kate's good idea" as a noun-phrase, it just needs the right context so it sounds natural.