Update:

Emailing the customer relations email got a response!

I received a reply from the same individual who had previously emailed me confirming me refund. Their claim was that: "when you made the reservation we charge 15.00 service fee per passenger, this amount is non refundable"

It seems to me that this is in direct contradiction to the DOT requirement that fares be cancelled within 24 hours "without penalty". The phone booking fee was quite clearly a part of the ticket process (even though it was not clearly disclosed to me over the phone), and the entire ticket (with fee) was charged to my credit card in a single transaction. This is similar to AeroMexico's own webpage regarding the 24 hour requirement.

I've clarified as such in my reply (including references to the CFR code) and will see what their response is.
Any thoughts here - if they do not resolve this, am I better:
- Complaining to the DOT
- Initiating a credit card charge back

Or both?