Just sorting this out for my 2009 K1300GT (yes, I know it isn't a Hexhead); DVLA shows it as Vehicle Type Approval = M1. This is a 4 wheel vehicle seating up to 8 passengers!
The correct classification is L3 as noted above.
Web page says post the V5C plus covering letter and wait 3-4 weeks.
Instead I used their E-mail an Enquiry
page - all went well, entered my details, reg nu, clicked NEXT or whatever thinking it would ask me what is wrong on the V5C.....instead it said "thank you for the enquiry, see email.
The email says "post your V5C, wait 3-4 weeks" etc. i.e. full circle.
BUT at the bottom of the email is a link to "reply to your enquiry"; use that and finally you can set out your issue, complete with vehicle details and the error.
Here's hoping this works without more DVLA customer discombobulation.
PS In the meantime, I contacted TfL via their "Make an Enquiry"
page. Explained situation. Got reply in 48 hours with screenshot of formal letter stating bike is Congestion Charge Exempt. Result.
HTH.