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It may be six or half a dozen unless you have been given a particular reason to go one way or the other. If the third party insurer is offering to pay all your losses in a way that is satisfactory to you then there is no reason to trouble your own insurer. However, if you go through your own insurer, as long as you recover your uninsured excess from the third party insurer you won't lose any no claim bonus you have and your insurer will recover its outlay from the third party insurer under what are called the insurer's subrogation rights. The end effect on you should be the same but I am happy to discuss further if you have have any additional information that might help.
If you think that they may decline insurance for any reason then yes, you should.
The third party insurer won't be interested in this and in any event won't have access to your original proposal form.