Pedantic point: 1975, not 1974:
https://www.legislation.gov.uk/ukpga/1975/63
Under Section 1(1) of that Act, the adopted son "may apply to the court for an order under section 2 of this Act on the ground that the disposition of the deceased’s estate effected by his will . . . is not such as to make reasonable financial provision for the applicant".
Section 1(2)(b), 'reasonable financial provision' is defined as "such financial provision as it would be reasonable in all the circumstances of the case for the applicant to receive for his maintenance".
The factors which the court have to consider, when deciding to make an order varying the terms of the will (and, where appropriate, the way in which such terms shall be varied) are defined within Section 3 of the Act. (See link above).
As I see it, it's by no means certain that a court would make an order to vary the terms of the will but (as your solicitor has already pointed out) it could still be expensive to get them to decide not to do so. It might still be best to make a
small offer, in order to get the claimant off your back.
[NB: Barmaid is our real expert here, as she's a barrister specialising in such matters. If she posts on this thread, her advice should be taken as more important than that offered by anyone else here].