For a bank note, you should use the numeral spelling "two hundred dollars".
The "I gave him two hundreds" is considering a "hundred" as a name for a 100 dollar bill, and that would imply you paid with 2 100 dollar bills, that is, not 4 50 dollar bills.
"two hundreds" can be used when a hundred X is a conventional size for a set of X's. For instance, when talking about white pape