Phippidus regius? In New Mexico?
I was just in San Miguel, Mora and Colfax Counties last week---Master Gardeners---so I re-familiarized myself with the 'lay of the land.' Excellent weather and travel conditions then---as opposed to a couple of weeks earlier with that 'bomb-cyclone.'
I understand your frustration with pictures, what you can see of each species and any notes on species distributions. All of my 'easy-to-access' sources are highly technical or just lack the fine details that might separate the 2 species once and for all. From what I can see, colors and patterns vary in different parts of the country for both species. Regius does seem to be found in the 'Southeastern US' which, I know, would include the Gulf Coast states, but how much of Texas is involved here is one question I have. I know we have audax in NM; our spider specialist retired several years ago----to Washington state---but he was interested in the complexities of that whole genus and confirmed, several times, that audax was in NM. From what I can see in 'bugguide', one would need a 'scope and a dead/preserved specimen for either dissection or ultra-close-up examination of critical structures on the body to separate regius from audax and other related species.
I would send you the fact sheet on audax that I found on bugguide except I am restricted to attaching only pictures, not documents. You can see what I can see, at any rate. I think you'll get that 'page' just by entering bugguide into your browser. Once you get that, there is a little search engine at the top right. Plug in 'Phidippus audax' or 'Phidippus regius' and you'll get series of pictures for each. Select one of this pictures---put your cursor on it and press 'enter.' Then see if the identification is the same as what you just entered into the search engine. Just above the picture are some tabs. Press the 'Info' tab and you'll get more information about that particular species. Neither gives distributions for these spider species, but you can get an idea of variation throughout the range of each species and other tidbits of information. Towards the end of the fact sheet are photos of the diagnostic structures used to distinguish one particular species from another.
This is a 'wandering answer' to your question but about the best I can do. Spider identification can be fairly difficult. Species differences are subtle; specialists rely more on physical features than colors and patterns---and none of those physical features are easy to see or find when specimens are so small---just a few millimeters. More recently, spider specialists are using additional data to validate species concepts based on physical features. Genetic analysis is one tool, as is mating behavior.
Bottom line: I would go for audax---still an interesting and beautiful---in its own way---little creature.