Texas is granted independence, and decides on a system where each county has equal representation. Conservative rural Texan voters either push for a withdrawal (if UN recognition succeeds) or prevent it from applying in the first place.
First, an independent Texas would never go for an equal county thing, even for its Senate. What would be far more likely is for the new nation to be divided into 5-6ish states based on culture, economics, politics, ethnicity.
It'd be something like:
West Texas and Panhandle - Midland or Amarillo as capital
Border counties - El Paso or Laredo
North Central Texas - McKinney, Dallas, Fort Worth
Hill Country - San Antonio (Austin becomes national capital)
East Texas - Tyler or Nacadoches
Gulf Coast - Houston or Galveston
If a secession were allowed, then it'd still be very likely to join the UN simply to not be Taiwan or a pariah. The business community would never allow it to not be an international business player. No UN membership = bad things for business.