Is there something special going on with "I've" in regard of Data's inability to use contractions?

You've already accounted for the actor slipping and the crew not catching it. So this answer simply looks for ways to provide potential in-universe answers.

But using a contraction in the next sentence after saying that he cannot do that?

Pedantically, Data did not state that he cannot (i.e. will never be able to) do that. He said that he has never mastered it. That means he does not exclude the possibility of learning it now or in the future - he only expresses that in the past (= up until now) he has not managed to learn it.

As long as you don't find proof of Data using contractions before the dialog around 24:00, then it's not a contradiction.

Together with Guinan, Data was one of the first to point out and observe Lal's contraction. This means he already knows what a contraction is, he just never used them. Seeing Lal use them may have given him the understanding he needed on how to use contraction in normal speech patterns, and adapted it accordingly.

In short words: android monkey see, android monkey do. During the runtime of TNG, we see many cases of Data mimicking or emulating human behaviors he does not yet intuitively understand (generally to comic effect). He could be doing the same with the contractions.

There are numerous mentions of how Data processes information significantly faster than the flow of a normal human conversation.
Similar to how a software developer can sometimes observe an error and make a pretty good guess at the source of the bug that causes it before looking at the code (as a developer, I can attest to that), Data could similarly process this observation on Lal using contractions, theorize about how Lal was able to quickly (since "birth") figure out how to use them, and learn from it himself.

If Data processes data usually fast, it's not impossible that to us slow-brained humans Data can change inbetween two sentences.

Especially since Data presumably knew Lal's brain inside out before he switched her on, and since Lal is not that old (since being switched on) yet, it stands to reason that the "contraction epiphany" is not far removed from Lal's initial state of mind - making it possible for Data to bridge this gap very quickly when observing Lal already doing it. He knows the start (Lal's brain at birth), the knows the end (using contractions as shown by Lal), so might be able to connect those dots more easily now that he's seen another android doing it.

Note also that you've only caught him using "I've", the exact contraction Lal used. I can use this to further suggest that Data may not have fully integrated contractions into his speech pattern yet, but has managed to integrate the one he explicitly observed Lal using.

Given that this is not explicitly addressed that Data could've learned from Lal, it's likely not what the showrunners intended. But based on the information provided in the question here, it's a plausible explanation nonetheless.