Third party insurance companies requesting that they do the work is common practice in the industry, as it keeps costs down. They would use their own authorised bodyshop, as the repairer and there would be a high probability, it is the same authorised bodyshop your own insurer uses anyway, so the work would be carried out to a equal high standard, as if you were going through your own insurer.
The problem in this case. is that you have already notified your own insurer Direct Line, so they are aware of a non fault accident. However if this is your own car, you could always speak with your insurer and request the third party insurer carry out the work but, it will most likely still be logged as a non fault accident, so you are not loosing anything other than the £200 good wil gesture from the third party insurer.
If, as I suspect, it is a Motability car, things get a little more difficult because not only notified DLM who will insist along with Motability, that the work is carried out by them and a claim made agains’t the third party insurer. If you had not notified DLM, you could have got away with it.
In this case, you will just have to explain to the third party insurer that it is a Motability car and you need to go through Motability’s insurance, as they are aware of the accident.