Does 2Checkout support token billing? Yes, and it works. However, there are some quirks with it:
You have to turn Demo Mode off on the dashboard or the 2Checkout and OmniPay API doesn't seem to work properly.
You have to send a Billing Address in order for the transactions to go through. If that doesn't work for you, such as for digital downloads, then you'll want to consider another payment gateway.
The billing address must contain a customer email and cannot be empty. Again, if that doesn't work well in your sales workflow, then you'll want to consider another payment gateway.
The billing address must contain a phone number, but CAN be empty.
I discuss this here as well as provide an OmniPay sample code snippet: https://stackoverflow.com/a/36807292/105539
Take a look at my project, TokenPay. I've weeded through the confusing docs and got it working on 2CO.
P.S. Don't ever forget your sandbox password -- they don't tell you this in the docs, but found out via tech support that their password reset on the sandbox login doesn't look disabled, but is disabled. (It would have been nice if they made that quite clear.) Instead, you'll have to create another sandbox account entirely when you forget a password there.