How to make Drupal Commerce tokens available in the Commerce Message module?
Even though your question is about the "Commerce Message" module, at its core your question relates to the Message module. And since you are using the Rules module, you can make the missing tokens (= the commerce standard tokens, as in your question ...) available by using the Replacement patterns for each message type.

So the answer to your question is "Whatever token you're missing, just make it available via Rules, by creating a replacement token for them". Refer to the answer to What's the purpose of "Replacement tokens" of a Message type? for way more details on that.

These Replacement tokens are really a hidden gem about how Rules integrates with Message. For an example to see them at work, refer to my answer to "Which user related token(s) can be used to create a Rule for setting up a message type?".