This is not intended to be a poke. Since text doesn't convey inflection of voice & such, I wanted to prequel with that.
I also understand your frustrations that after a decade the DPS is asking for something that, when you initially disclosed it on your initial application, was deemed as not disqualifying. I don't have insight into their adjudication process, but perhaps some new process/procedures have been implemented. If so, I'd imagine there's a backstory as to why they're now asking such questions. ((That is NOT to imply there's a reason for YOU SPECIFICALLY, just in general that if someone else that had been approved for an LTC in the past slipped thru the cracks that may be the reason for updated processes/procedures. I was quick to disappointment with DoD when it came to light that the crazed loony tune active shooter SHOULD have been a disqualified person, but the Air Force failed to send his domestic violence disqualifier to NICS. ))
I also respectfully disagree that it's capricious nor is it arbitrary, as the DPS is required by code to conduct a background check for every renewal. If the DPS is asking for this now, I'm thinking in good faith it is something they need to close the loop & get your renewal either approved/ disapproved. I see that as doing their job, according to what they're charged & authorized with doing in Sec 411.176, not over reach or exceeding their authority but rather doing what they're actually required & authorized to do.
That is the subparagraph that immediately precedes the one you cited above, & clearly establishes their legal authority/ability to ask for follow up documentation."(d) On receipt at the department's Austin headquarters of the application materials and the result of the investigation by the director's designee, the department shall conduct any further record check or investigation the department determines is necessary if a question exists with respect to the accuracy of the application materials or the eligibility of the applicant, except that the department shall complete the record check and investigation not later than the 180th day after the date the department receives the application materials from the applicant."
Reading a little further down in the code, in Sec 411.180 you'll find your legal recourse if you remain determined to not send the email that Infoman mentioned above. Once the DPS denies your renewal they will send you a denial letter stating as such ((that's assuming they don't just cancel your application for renewal as "incomplete" for refusing to respond; you typically have a specified time limit to respond before they cancel the application To mitigate that, perhaps do respond respectfully stating your position, & requesting that they send you a formal disapproval letter if they aren't going to renew your LTC)). You'll have 30 calendar days from the date on the disapproval letter for the DPS to receive your request for an administrative hearing. The DPS will coordinate with a Justice of the Peace or Magistrate local to you, & that justice will serve as the mediator. You will be able to explain why you think the DPS got it wrong, & the DPS will be able to send a representative to explain why they denied your renewal. That judge will make a determination, & if you lose out you can appeal it. On appeal, whatever is decided is decided.
Since the DPS isn't (from my reading of the code) asking for something unreasonable to complete the background check, I honestly wouldn't think a JP would judge in your favor on this argument.
The solution I see is to send what they're asking for & receive your LTC ((if all checks out)). And if not, you can use the due process laid out in the code, or revert to constitutional carry.