This. And this again.
Both pugs and French bulldogs are snub-nosed breeds - many/most airlines will not transport them in the cargo hold these days,and with good reason. Take them along as carry-on in the cabin if at all possible.
Both Air France (AF MLA-CDG-ATL plus Delta ATL-TUS) ) and Lufthansa (LH MAL-MUC-ORD plus United ORD-TUS) will transport your dogs in the cabin (provided they don't weigh more than 8kgs including carrier) and fly between MLA and TUS.
You will need to have a companion as in-cabin pets are limited to one per passenger on all airlines in question.
I would book directly with the airline and not use a travel agent. Experience has shown that if changes have to be made to accommodate pet-unfriendly aircraft swaps etc., it's best to not have a middleman 'owning' the ticket.
Book your connections with enough time to allow for getting the dog(s) checked in with the connecting airline at the transit airport .You will need to book the pets separately with each airline involved in the itinerary (and pay the fee to each airline separately as well).
Be aware of the CDC-mandated restrictions regarding the importation of dogs to the USA. Malta is currently not on the list of 'high-risk" countries so you should be good.