I think your employer probably has all the information about exactly what they can and can't do. They can certainly sponsor you for an Essential Skills visa.
https://www.immigration.govt.nz/empl...-accreditation I don't know if their accreditation will avoid the labour market checks. But anyway, if your job is on the LTSSL list and you meet the requirements for it, that WOULD mean no check.
https://www.immigration.govt.nz/new-...-visa#criteria (See under Evidence for the
Availability of New Zealanders section.) I can't find "transport operator" in the skills checker, or ANZSCO, under that name. Are you a driver, or do you organize?
If you succeed in getting another work visa, you will be eligible to sponsor your partner for another visa, as long as he "held a visa on 28 August 2017, granted on the basis of their relationship with you" - see that regulation that you quoted yourself.