Upvote:6
The I-571 Refugee Travel Document is perfectly sufficient for re-entering the US. You should have no trouble with it unless you do something that makes the US question your status as an asylee... such as using the passport of the country you claimed asylum from.
It is a very, very bad idea to continue to use that passport in a way that the US might find out about it. If the US learns you are still using it, your asylee status could be revoked. This also means it's a very bad idea to have the Mexican visa placed in it. The details of everyone's departure from the US by air is passed to the government by the airlines; this is how the US does not have exit immigration controls as most countries do. If your departure is recorded on the passport of the country you claimed asylum from, your status is at risk.
You should apply for the Mexican visa with your Refugee Travel Document instead. This is perfectly safe and you do not have to worry about entering Mexico or returning to the US.