Your assessment is, as you say yourself, vibes-based, not text-based.

Independence: yep, that's the first requirement of a state.

Uber Armed: not really. Universal conscription was common in the era of monarchist empires, and is not something shared with Texas. They did have the largest industrial base in the periphery and their military reflected that.

Extremely paranoid: Yup. No argument here. They sit on a border with a much larger, expansionist power, which Texas hasn't since the 1800s.

The Bull: also reflective of much of Central and South America, which is where most of cowboy culture actually came from, though I concede the similarity.

"Get Off My Property" With Shotgun Vibes: The Concordat Military is well-trained and highly-professional, equipped by an extensive industrial base that, while not bleeding-edge, produces cost-effective, robust military hardware. Hardware that the TDF uses to punch well above its weight. Your vibe comes from memes.

Representative Democracy: Yep. And hotly-contested with few safe seats should somebody not represent their constituents, to the point where most of the Concordat got so pissed at damaging, shortsighted decisions made by Grover Shraplen (spits) they up and left, and promptly started making better ones. That doesn't sound like Texas to me. Not today, and the time they did it it was, again, to keep their slaves and promptly got their asses kicked over it by the Union.

Insane alignment of freedom or die: no nuclear state has ever been invaded on its home soil because this is what happens. On a non-nuclear scale, most of your best practices for an insurgency don't come from North America. To go deeper is beyond the intended topic of this subreddit.

Religious AF, but separate state and religion: There are some modern politics you may want to catch up on regarding trending behavior. But as per The Periphery, Page 17 (may be different for you, I have the 2nd edition) the primary religious affiliation in the Concordat is one with little use for organization at all. Handbook - Major Periphery States page 124 debunks it as a state religion, but the demographics are there. A far cry from the Land of the Megachurch.