Moving companies charge by weight plus additional fees like fuel surcharge, elevator fee, shuttle fee (if the big truck on either end can’t access your place, everything is loaded into a small truck first). Not Halifax but for some reference, I moved a two bedroom from Toronto to Vancouver and it was about 6k but shuttles were required on both ends and a storage fee for a week.

I used Great Canadian Van Lines and they were great. Most of the moving companies can give you a rough quote online or through a follow up call.

There’s also a federal tax credit for moving for a job that you might want to look into. The 4K from the company might grind against it but it could cover the additional amount.