Via El Paso, I would advise taking TransMountain & the Loop (TX 375),
If towing it is better to take NM 404 east toward Chaparrel at exit 162 over to NM 213 then south to US 54 west to get to Loop 375 south. No steep climb up or down and the mileage is about the same, with less traffic.
Transmountain with a tow is no good, you would be better to stay on I-10 to catch 180 in El Paso and deal with a few stop lights on the way out of town.
If the weather is fine use US 70 and 82. The down slope from Cloudcroft is an easy, gentle, scenic drive. In fact going via 70/82 is much more entertaining than 180 if the weather cooperates.