Dispute with relocation company

I recently relocated to US from UK. I wasn't sure if I was going to NYC or DC so I asked the shipping compnay what the difference in charges would be if I ship it to DC but later have to redirect it to NYC, they told me GBP 50.

After they pack my things and load them up to the ship, they told me they made a mistake quoting me the difference in charge. I ask them to requote, they did not. When I asked for delivery to NYC, they gave me a quote of $4,000.

I dispute the charges and they blame their UK partner, saying what was quoted was between their UK partner and I, not the US affiliate. They would not deliver my things unless the $4000 fee is paid.

What do I do? I already went through BBB mediation, nothing resulted from the mediation.