In Texas, there is no such thing as "not a terminable offense." Texas is an employment at will state. Typically, unless an employee has an employment contract, or is employed under a collective bargaining agreement through a union, the employer can modify or terminate the employment at any time with or without cause. If an employer, at any time, decides they no longer want to employ someone, for any non-discriminatory reason, that employee can legally be terminated. However, an employer generally cannot modify or terminate the employment for prohibited discriminatory reasons (such as racial discrimination), or in retaliation for certain protected actions (such as whistle-blowing). The situation you describe does not appear to constitute prohibited discrimination or retaliation.
Your question has been answered as a courtesy. This is not paid legal advice. Nothing in this communication is intended to create an attorney-client relationship. Unless expressly stated otherwise, nothing contained in this message should be construed as a digital or electronic signature, nor is it intended to reflect an intention to make an agreement by electronic means.