I had a look at the 1988 Road Traffic Act which governs yir offences and I can�t see how yir solicitor thinks having no MOT is related to the Due Care offence since the Act says
" 3. Careless, and inconsiderate, driving. If a person drives a mechanically propelled vehicle on a road or other public place without due care and attention, or without reasonable consideration for other persons using the road or place, he is guilty of an offence."
Having no MOT is a separate offence.
On the Association of Natiional Driver Improvement Scheme Providers website here,
http://www.driver-improvement.org.uk/ndis_f_a_ q.htm it says in reply to
�What are the criteria for being offered the course?�
The decision to offer a course is based upon sufficient evidence available to follow proceedings for �driving without due care and attention or reasonable consideration to other road users� contrary to section 3 of the road traffic act of 1988.
In addition
The driver must have a full licence.
There must not be any other offences to be dealt with by prosecution at the same time as the due care offence (i.e. no insurance).
The offer must not be made within 3 years of the completion of a previous Driver Improvement Course. "
It seems then you are out of luck, since you have the two offences being dealt with at the same time.