If you genuinely believe you are in the right, and they are in the wrong, don't pay it. Let them instigate court proceedings (they will file the dispute with the court, you will then receive a court summons which is an invitation to attend a public court hearing, in front of a judge, with them present). This will probably be a small claims court hearing. As for bailiffs, this is far too early for that.
The debt, and the dispute, can't be ignored by you. I suggest that if you have confidence that you are right, prepare to attend court. If you can't pay the charge, be prepared to attend court also. But be aware the judge could rule either way, so make sure you have a solid case for the ruling to go in your favor (otherwise you'll no doubt have even more costs to pay out). Well prepared documentation for the judge goes a long way - gives them time to review the case before it even reaches the court room. Do not walk into court unprepared.
What ever you decide, good luck. It is stress, and it is hassle, I agree, but if you let people walk over you, they will
walk over you (one of life's lessons).
If it helps, here's a little story for you. Some years ago now, my dad needed a contract written up by some solicitors. In a meeting, they both verbally agreed that this would cost "about £60" (this happened in Scotland, btw). My dad got billed £120, twice the amount. He disputed it. It went all the way to a small claims hearing. My dad chose to represent himself, and walked into the court room. The legal firm had 10 of their best barristers in there, with their wigs and gowns, all ready to completely destroy my dad (legal intimidation, at its best). But he calmly entered, presented well prepared documentation to the judge, and the hearing was over in less than 5 minutes. The judge basically said to the legal firm "there is no case here, stop wasting my time, £60 is what was agreed". The firm ended up having to pay all my dad's expenses too.
He knew he was right. This was a clear example of a company trying to screw over the individual, as I'm sure they have done before, and as I'm sure happens every day, except most people do not have the time or confidence to rebut - they just accept it (reminds me of this country's current political state and government, but let's not go there). If you know you are right, and you can prove you are right, the judge will most likely rule in your favour, because you are right