Under Texas Penal Code section 46.01, Firearm is defined as:
any device designed, made, or adapted to expel a projectile through a barrel by using the energy generated by an explosion or burning substance or any device readily convertible to that use. Firearm does not include a firearm that may have, as an integral part, a folding knife blade or other characteristics of weapons made illegal by this chapter and that is:
(A) an antique or curio firearm manufactured before 1899; or
(B) a replica of an antique or curio firearm manufactured before 1899, but only if the replica does not use rim fire or center fire ammunition.
(4) "Firearm silencer" means any device designed, made, or adapted to muffle the report of a firearm.
Tex. Penal Code § 46.04 makes it unlawful for a convicted felon to possess a firearm. There is nothing in that section that references a BB gun or an air rifle. Nor are there any cases that reference that code section and air rifle or BB gun.
See United States v. Rearick, 181 Fed. Appx. 303, which stated:
Rearick's status as a convicted felon would not, however, preclude his possession of the two BB guns. U.S.S.G. § 1B1.1 ("A weapon, commonly known as a "BB" or pellet gun . . . is a dangerous weapon but not firearm.").
I hope that helps. Please let me know if you have additional questions.