I'm looking for a quite simple bitcoin-based donation service that would keep track of the $USD sent to a particular address/wallet and display that on a private url page.
Don't you think looking at
https://blockchain.info/address/yourbitcoinaddress will be enough for what you need? Just click on bitcoin amount it will be converted into USD. They will display both total received amount and current balance in USD depending upon current bitcoin value.
Anyway it looks free and powered by the site coinwidget.com . Give it a try
Looks like this is enough but they will display total received amount in bitcoin rather than in USD like OP have asked.