Is "that've" a valid contraction for "that have"?

It's certainly found in speech alongside who've and which've, and that's how the pronunciations are normally represented in writing. It's a matter of judgement whether you use them in writing when not reporting actual speech, depending on the degree of formality of the context.

As a native speaker of English, I consider "that've" to be a perfectly valid word. Grammarians be damned!

I definitely use it in spoken English (native of the United Kingdom) and have heard it used often here. I've never seen it written that I can remember. Google books search doesn't find much and the Google Ngram search only brings up a very small number.

I do use it in writing. But with a caveat. I write novels and only use it in dialogue to simulate local speech. (Pacific Northwest for me)

My editors accept it in these circumstances. And only these circumstances.