I was wondering if anyone knew if I could send BTC to an Exchange or wallet that supports BCH/BTG and get the coin equivalents credited to my wallet, or is it past the point where I can claim them?
Not possible. You must have the funds stored on that exchange while the fork is happening.
Better claim your forked coins via various methods available around. Of course you need your private key here.
I've used the
Coinomi method to claimed my BCH but nowadays I believed there are lots of easier ways to claimed forked coins. Just give it a search or you can try the above suggestion. Don't have experienced using it so read feedbacks for reference.