If you get an email confirming your pending deposit before you've got a confirmation you may also be able to double spend the original transaction allowing for you to mitigate paying extra fees (as the 2 transactions could be replaced by one).
This introduces additional risk without knowing how the back end of the exchange works. There are plenty of places I have sent bitcoin which give you an address and monitor that address for an incoming transaction. As soon as an incoming transaction is detected, they stop monitoring the address and just wait for one (or three, or six, or whatever) confirmations. If you replace your initial deposit with a new transaction, there is a chance that the new transaction is not detected by their system, and then OP will likely have to spend days or even weeks going through the exchange's support team (which are almost universally awful) to get his coins credited to his account. This would only be safe if you are absolutely certain the exchange only gives you a single deposit address and does not change it.
Similarly, if he sends a small amount prior to sending the larger amount and the exchange does change the deposit address, if both transactions are to the same address he will again be faced with the above scenario.
Although I've seen this behavior at payment processors, I didn't at the centralized exchanges. And at the exchange OP can double check if the address has change before attempting such a risky operation.
On the other hand, I always suggest use of RBF just because the fees can change in widely, and the exchanges wait for confirmation(s) before sending the mail. So the double spend will be no longer possible.
Also I've also seen in the past at least once an exchange suggesting (at least for an altcoin) to first send a small amount. It can be useful if more complicated things have to be set/checked (address and some other details too). But I don't see it useful for Bitcoin, where only the address is necessary. Still OP has to be careful to not mix up Bitcoin with BCash and such. All in all, if OP needs a peace of mind, he can do a small send first time, maybe in a week-end. Otherwise it's not necessary.
And yes, if anything goes wrong, the proper tool is the support and the proof (tx id). I myself faced such a problem, long ago. I've sent the deposit, it was not the first deposit I've sent there, but that one was not credited. I've contacted support and after a while (I don't remember if it was hours or days though) it was handled/fixed/credited.