Not on TD status. A TD spouse may live in the United States and study, but TD carries no employment authorization for any employer, including the principal's own client. She would need her own qualifying category.
TD is residence and study, never work
The TD classification exists to keep a family together, not to admit a second worker. A spouse admitted in TD may remain for the same period as the principal, may enrol in school or university, and may travel in and out of the country, but may not be employed, paid or self-employed anywhere in the United States. The engineering firm offering her a coordinator role does not change that; the offer would have to be assessed on its own against a listed profession, her own credential and her own citizenship. Children admitted in TD have the same limits and lose the status at twenty-one or on marriage. If the family's plan depends on two incomes, the honest planning step is to test the spouse's occupation against the USMCA list now, rather than discovering at the border that TD is all she can hold.
Hypothetical example: the surveyor's spouse is a dental-office administrator offered temporary remote scheduling work for the U.S. client. TD does not supply work permission merely because the tasks are remote, part-time or paid by a Canadian payroll company. A first review should identify the location where duties are performed, the paying entity, the intended classification and whether the spouse has an independent route. The offer letter and payroll arrangement settle the practical question; calling the role volunteer help when it is productive paid work creates a credibility problem. Sequence the family's admission records after the principal's TN package is ready, and carry proof of the relationship and the principal's current admission. If two incomes are essential, resolve the spouse's category before signing a lease or promising a start date.