If you are palmed off by the council with things like "it's in the hands of bailiffs", the law does not obligate you to trade with a third-party.

Invite the council to contact you again when the bailiff has returned the liability order. When the bailiff has returned the case to the council, you are no longer liable for the £310 fees

If you have a debt, then get it paid direct to the council to extinguish the enforcement power, or take steps to protect yourself

Bailiffs collecting council tax act under an enforcement power called a liability order. It does not empower a bailiff to break and enter property, "get a locksmith", or use force against people. Provided the door is locked shut, and there is no vehicle of yours outside, you are safe. The bailiff returns the case back to town hall administration after 90 days if he is unable to recover the money or take control of goods.