Landlord Demands Certain Moving Companies

Tell them then you are staying, and not paying rent, and in 6 months or whatever they can explain to a judge how they wouldn't let your bonded and insured movers into the building to move you out so you were forced to stay and effectively held hostage.

I would only do what 30 says AFTER presenting them the paperwork of a properly-licensed, bonded, and insured moving company, preferably the one that moved you in. Then demand that the rejection of this company be given in writing.

Are you in a rental building, condo, or coop? In a condo or coop, your lease probably says you'll abide by the building rules, which may have been duly changed to restrict movers who enter the building. In a rental building, the lease is probably silent on the matter, so your case is stronger.

It's a rental. The problem is I'm signing a lease with a new landlord and will owe rent to them. If I can't move it will be a problem.