Per this unofficial source,
Texas exempts fifth wheel hitches, but no mention of gooseneck balls in that link.
However, the actual code exempts "fifth wheel and gooseneck semitrailers" towed by a passenger car or light truck. You'd have to dig deeper to determine the definition of a semitrailer, but it's probably any 5er or gooseneck trailer, not a "bumper pull" trailer.
Here's the actual code:
Originally Posted by Texas Revised Code
Safety Chains – TRC 545.410
(a) A person may not operate a passenger car or light truck while towing a trailer, semitrailer, house trailer, or another motor vehicle on a public highway unless safety chains of a type approved by the department are attached in a manner approved by the department from the trailer, semitrailer, house trailer, or drawn motor vehicle to the towing vehicle.
(1) does not apply to trailers, or semitrailers, used for agricultural purposes.
(2) does not apply to trailers, semitrailers, house trailers, or drawn motor vehicles operated in compliance with FMCSR.
(3) does not apply to trailers, semitrailers, house trailers, or drawn motor vehicles which are equipped with safety chains installed by the original manufacture before the effective date of this section.
(4) does not apply to fifth wheel or gooseneck semitrailers.