This isn't really an issue of succession laws, though. British (and Commonwealth) succession laws limit the succession to the legitimate descendants of Sophia of Hanover. Legitimate isn't really expanded upon - thus, "legitimate" is restricted to what other laws define as legitimate.
The succession wouldn't necessarily need to be changed - you could simply change the rules regarding who the legal parents of a child born via a surrogacy are. In the current system, the legal mother of a child born via a surrogacy (in Britain) is the woman who gave birth to the child, and the legal father is the birth mother's husband or partner (this applies to same-sex relationships).
Looking it up, if the biological parent(s) get a parental order (which has to be applied for within 6 months of the birth), it might be that the child will then have the same legal status as a child born through natural means to them - some FAQs about this can be found
here, and the way I'm reading it there's certainly grounds for a child born via a surrogacy whose parents are established via a parental order could inherit a title.
If, however, a parental order isn't possible, then the child is adopted, and thus has the same legal status of an adopted child who isn't the biological child of the parents - that is, they can use the courtesy titles due to a daughter or a younger son, but aren't themselves able to inherit any titles.
I wouldn't expect this to be challenged by the BRF though - typically it seems that this kind of thing is challenged first by your average person, then members of the peerage, then finally once it's "accepted" among the majority it is seen in the BRF.