If the goal here is to get rehired - do nothing. You would be wasting your time. You weren't employed there long enough and weren't dismissed for an "automatically unfair" reason so you cannot challenge the dismissal.
It sucks, I get that - but there's simply no legal basis to challenge this on and you've already been through their internal processes (twice) which would be the only way to affect this. Essentially your short tenure (< 2years) and the lack of an "automatic" reason means that in an employment sense of the term this wasn't an "unfair dismissal", no matter how "unfairly" it may have been handled in the colloquial meaning of the word.
For your own sake as much as anything - Let it go.
If you wanted to go after the employee you feel wronged you then there maybe ways to do that (IANAL) but in that scenario you need to be talking to a solicitor, not internet strangers - if there needs to be any documentation requested from the employer's records to support that case your solicitor will direct you on what that would be (and would potentially even handle such a request for you themselves).
If you were interested in potentially working for that company again in the future at some point then a legal outcome proving that the other employee lied to get you sacked might help - although it's far from a certainty and it wouldn't entitle you to your old job back - that job is gone.