You should certainly take the evidence to the mediation. As for 'getting it in front of the court,' most of the time you would use this in mediation or while the spouse is on the stand to prove he is lying to the court... an attorney does this during cross examination and if hired, especially with the facts you provide, you could end up with quite a bit more than what the spouse is putting on the table now...
Disclaimer: This answer is intended for informational and educational purposes only and should not be considered legal advice or forming the attorney client relationship. Please consult a lawyer before taking any action.
If a bank was used in the transfer and you know which bank, you can subpoena the bank's records, which would show the transactions. It takes a MINIMUM of 24 days to get bank records, so if you are headed to mediation soon, you may not have those records. However, the issuance of the subpoena might give your spouse notice that you plan to investigate this carefully.
When you get the bank records, you'll introduce them in court as evidence and it will become evident that your spouse lied. There is no "motion to declare other party a liar." That's just a result of the court's (or jury's) interpretation of the evidence.
If you don't have an attorney, you need to hire one right away. Do not go to mediation alone. You must have an attorney.