Thanks all, and sorry for the delayed response.
I appreciate all your responses, and it is good to know that, generally, everyone is agreeing with getting the third party to deal with this.
Considering all your response, my research, and a cracking video by 44teeth, I am happy Churchill (third party) will deal with this.
In fairness to both sides, they have been reasonable, but Churchill have let me book my bike in at an official Honda dealership (it is a CB1100 btw, and damage has gone across the side of the bike, and possibly bent the front mudguard). The other (4d) were taking it away, booking engineers etc.
My main issue now is no-one wants me to ride it to the local dealership in case something is wrong as it is ~hours drive away. I think the bike is fine, but will probably speak to Churchill about getting the garage to pick it up. They have offered this, but I suspect it could be overkill considering the issues (I have sent damage photos). However, I suppose it will be far easier for them to claim any other things/damages I cannot see are (as I am not a professional) are due to me getting it to the dealership/the ride there. Also, I lack the skills to really look at the bike to ensure it is safe so getting a pick up does sound like a good option.
Thanks again everyone!