Is the proposed ownership a tenancy in common or a joint tenancy? A joint tenancy is when the two own the property but on the death of one the survivor becomes the owner of the whole property . That is automatic, the deceased's interest in the property ceases on their death, therefore there is nothing of it for them to leave to anyone.
A tenancy in common is when the two own the property in whatever share they decide.It might be half each or two-thirds to one third or any other share. Then both have an interest which they can sell or pass on on death.If one wants the whole property to be sold, to realise their share,and the other doesn't want it sold, the first can achieve that with a court order for sale .
Is it that the solicitor now wants the sole owner shown to be the person who is paying? It is not clear from what you say. If he/she wants something else that's more mysterious.