Your agreement should specify the currency in which you will be billing, as well as the bill-rate, which I assume it already does. Then that is the bill rate. If your agreed rate $50 US / hour, then that is what your client should expect to be billed and to pay. If it is 4000 <some other unit> / hour, then that should be the expectation. Exchange should not enter into it.
But of course the exchange rate does matter to you or your client when they have to purchase another currency to pay you and that rate changes significantly. By asking to roll back the agreed exchange rate to a prior date, your client is really asking you to accept payment in his currency instead of in yours, effectively lowering your bill rate. Would he do you the same favor if the difference were reversed?
In future contracts, or in this one if you're willing to renegotiate to keep the client happy, you might specify a referene exchange rate - perhaps its 2-year average - and agree to split the differences during the life of the contract by billing and accepting at a rate 1/2 way between that originally specified rate and the rate on the billing date. It may feel fairer to your client that you'd be willing to take 1/2 the risk (if you would be). Setting the reference rate amounts to exchange speculation which involves risks and I assume is not related in any way to the contracted work. One of you could still take a real loss, so you might want to set some exchange-rate limits around the reference rate that you both agree you won't exceed without stopping to renegotiate.