Should I register my novel's copyright before sending it out to an agent?
Short answer: No.
Longer answer: The book probably won't sell anyway, so that's $45 you'll never see again.
Even longer answer: Copyright exists automatically from the moment the work is first fixed in tangible form. The records you make in the course of doing your everyday business, your printouts, your rough drafts, provide more than adequate proof of your original composition.
Longer answer still: Publishers routinely copyright works in their authors' names. Breaking that routine slows them down and costs them. When a new book comes out with a copyright date that's some years earlier (and face it, if you sold your work tomorrow it probably wouldn't hit the shelves for a couple of years) readers in bookstores looking at that date would figure that the book was old, or a reprint. Many would put it back in search of something new.
Go ahead, copyright your book if you have money to burn and can't get to sleep otherwise, but understand that you're wasting your time and money. There is no market for pirated slush. None at all.
Among agents there are two basic kinds: Honest and dishonest. Honest agents aren't going to pirate your work because they don't just want this book, they want your next, and your next, and your next.... Someone who can write a publishable manuscript is rare enough that they aren't going to throw him or her away for a one-shot advantage, and if a book is successful the odds that you wouldn't learn of it approach zero.
A dishonest agent isn't going to pirate the book either, because they couldn't sell a book, even a publishable one, if you held a gun to their head. How are they going to sell a pirated work? Their source of income lies in the fees they collect from writers. Plus, again, if the book has any kind of success, you're certain to find out, and their cheese will be in the slicer for sure then.
An honest publisher isn't going to buy a pirated manuscript because, not only they are honest, but they're going to want to work with the writer to improve the work. No one but the original author could possibly do that.
A dishonest publisher isn't going to "buy" a pirated work because their business depends on the author himself buying multiple copies of his own book to peddle at flea markets. Who's going to have so much ego invested in a manuscript they stole to pay thousands of dollars to pretend to be its author and go from bookstore to bookstore begging the managers to carry a copy?