He can't have met her that day. He just can't. I don't believe that. <--- This is a problem because you are using current tense (present) for something that happened in the past. Big no no. He couldn't meet her that day, his wheelchair was broken. <--- OK, but I would make that a semi-colon instead of a comma because the idea is not a directly stepping down from meeting and the wheel chair being broken. The two simply have nothing in common. He couldn't have met her that day without a wheelchair. <--- OK. Remember, they all mean different things. Although if you are asking about grammar, then I've pointed that out for you.

He can't have met her that day. He just can't. I don't believe that. < NO replace 'can't with couldn't' in both places He couldn't meet her that day, his wheelchair was broken. < OK, I would lose the comma and add the word BECAUSE in its place. He couldn't have met her that day without a wheelchair. < GOOD

All are correct. The first case is tricky and you got it right. When using modal verbs to write about the past the rule is can't + have + past participle. I suspect you may already know that. :)