Ultimately it's your call what information you provide, it's your data after all, but some considerations...
Group Nexus do need to verify that you are who you say you are, so that they don't wrongly hand out someone's personal data. In theory the way they do that is by comparing the data you send them with the data they have on file (i.e. name and address). Given that they don't currently have your date of birth, it's not clear how you providing it to them helps them to verify that you are the data subject in question, so it would seem they shouldn't be asking for it (there's some guidance on this
HERE - as an aside you might want to let the ICO know that Group Nexus are requesting unnecessary personal data before complying with SARs).
That said, if they're being stubborn and it gets in the way of you getting a response to your SAR, which is the main aim, then you may decide to provide some of the data. Be careful that none of what they are requesting could harm your case (if they are asking who was driving on the form, for example, this definitely shouldn't be provided).
