Ask for the start date to move, or for a documented temporary permit from the state board. Do not present at the border with a conditional offer and an incomplete licence. TN admission is granted for a period of up to three years, so a later start costs little compared with a refusal on record.
Move the date, not the evidence
The instinct is to travel on the promised date and sort the licence out afterwards, and that can undermine the application. An officer at a port of entry decides on what is in front of him, and a start date that cannot lawfully be met undermines the claim of prearranged professional employment. Hypothetical example: a Lake Country nurse's endorsement application shows a six-week estimate and her start date is in four; she asks the hospital to place her in the next orientation cohort, obtains a reissued letter with the later date, and applies once the licence is issued.
If the hospital cannot move, ask the state board whether an interim permit exists and whether the hospital accepts it. A refused application becomes part of the immigration record and may be considered in later applications or admissions, while an admission granted two weeks late is simply an admission. Sequencing costs time; skipping it can cost the case.