Am I legally allowed to accept fiat currency online and return a counterparty asset/token? I want to sell the tokens (counterparty asset) for my online game via standard shopping cart. Is that "legal" or is my only option to trade BTC for the token via exchange?
It depends on where you live, where you sell and who the buyer is.

If your jurisdiction regulates this stuff or your buyer is say, a US citizen, regulations may apply. You may be able to work around that by implementing certain basic rules such KYC/AML, either full blown, or basic (such as the user confirming they're not a US resident or citizen, etc.)

There are several games that do exactly that on the Counterparty platform without issues, but that doesn't mean anyone can do it.

Recently there's been a growing number of regulations about ICOs and "tokens", so best check with someone who understands laws and regulations that apply to your territory and is qualified to give legal advice.