Interest rate parity is not sufficient now
There is a cross currency basis between the IRP result and observed market prices, because essentially Libor does not represent the cost of funding; in particular the implied USD funding rate deviates significantly from the USD Libor.
Cross currency basis (as a swap) is a traded quantity which covers that difference.
Yes, it will change, and the difference will be more apparent
Using the overnight rates (whether the current ones or the new reference rates like SOFR) the basis is even more apparent, particularly when there is a lack of liquidity in the USD market. Attempting to use IRP to price FX is even worse in this scenario.
The rates are already available, so you can try the experiment now; calculate the implied rates and compare to market rates. Again USD is the most indicative.
The basis is not straightforward, not static, and varies over the reporting cycle
Many such kinds of basis are largely static, but the cross currency basis is blown by many winds, including reporting cycles, credit spreads, and so on. The FX forward market is used extensively by institutions to fund their need for currencies for reporting, which is particularly apparent with USD where they do not have access to overnight rates. As a result, the no-arbitrage conditions on which interest rate parity is based do not apply, and the effective funding rates deviate from the overnight rates strongly.
Decide whether it matters to your client
Perhaps they will not mind a gap between their implied FX and the market prices; evidently they do not mind at present. I would suggest analysing the basis from the overnight rates and looking at how much such a basis would move the actual calculated prices. If they do not move them by enough for the client to care (e.g. if the spreads on prices are wide and the maturities are short), then they may be able to avoid making any significant changes (beyond repointing at different swaps) for now.
It may not surprise you to know that getting the calculation right is the subject of a lot of proprietary analysis and 3rd party software :)