TN requires prearranged professional employment, and a conditional offer may still describe real employment. Whether it is sound to proceed while the condition is outstanding depends on the wording of the offer and the timing, which is a question for counsel on the specific facts.
Look at what the offer commits the employer to
An offer that names the profession, the duties, the supervisor, the location, and the compensation, subject to a clearance, reads differently from one that reserves the right to withdraw for any reason. Ask counsel to read the actual language rather than a summary, and to advise whether the case is better prepared now and presented later, or held until the condition is resolved.
The profession’s substantive qualification still needs assessment independently of the spelling or name history. The record that decides this is the employer's offer letter, read together with the job posting and any commission plan. Regardless of cost, eligibility still turns on Canadian or Mexican citizenship, prearranged work in a listed USMCA profession, and that profession’s own qualifications. Do not rewrite the duties to match; the officer will ask about the real work, and the applicant will have to describe it.