"That that" spoken are two different sounding words so it makes sense. When it goes from verbal to written and I see it, I will almost always try to rephrase things to avoid that combo. It just jumps out as totally wrong.
I think it depends on if you want to emphasize something specifically or not. Second 'that' is the default it seems, but I first expected 'was' to be emphasized in this sentence
In fluent speech, the conjunction (the first "that") is unstressed, and as a result some speakers reduce the vowel a bit toward schwa. However, if you told those speakers to carefully pronounce each word, I bet they would pronounce the conjunction and the pronoun the exact same same. A more common example of this kind of reduction is the word "to", which is almost always reduced to /tə/ ([tə] ~ [tʊ] ~ [ɾə] depending on dialect and surrounding words) in everyday speech when unstressed.
Fun fact, you can reduce just about every unstressed vowel in English to schwa (if it's not already a schwa) and still be largely understood.