The best time to prune was 2 months ago. If you are going to do it do it now. Prune ONLY the stuff that is interfering with your gate. As you said, that tree can ill afford to lose what it has expended energy and resources to make. Your pruning cuts may make it more susceptible to sunburn and wind damage so EVALUATE that when deciding on the cut(s). Make them thinning cuts so grow back is less likely.

Here is a link that might be useful: Pruning Deciduous Shade Trees

The Desert Botanical Garden has done research that shows desert trees (mesquite, palo verde, acacia, etc.) are best pruned in early May. They recover more quickly at that time. Of course follow the cardinal rule and don't remove more than 25 percent of the living tissue (leaves, branches) in any year. You can do light trimming/shaping nearly anytime.

You could use a 'reduction cut' for the lower branches which would reduce their length, but leave them functioning for the tree for another year or so before removing them completely. These temporary branches will help produce stronger trunks with larger caliper and better root systems. Trees with these temporary branches grow faster than those without them. (Ed Gilman, University of Florida)

Here is a link that might be useful: Reduction cuts