Just a point, a solicitors letter has no more legal standing than you just writing it yourself. So save the £50 next time and just use recorded delivery.
As to the problem I can't offer much help, sorry. The solicitor is correct in that he has to pay his way.
If his partner ( I assume they are not married?) owns 1/3rd then he needs her permission to sell, as I'm sure you know.
You will have to get back to the solicitor for advice. ( but write any letters yourself) The partner may be eligible for legal aid , not sure now as the rules have been changed, but get her to apply anyway , they can only say no.
The sooner she tries for legal aid the better as it can ONLY be awarded to one side in a 'matrimonial' dispute,( which includes partners & not married couples only )
Are there children involved? that could swing it for legal aid if there are!
Partner should also contact 'Shelter' the homeless charity for help as there is the possibility of losing her home.
Now I think of it, Shelter may well be more help than a solicitor !!
Shelter website here ,they have a telephone help line that she needs to use!
http://england.shelter.org.uk/?_ga=1.257794955.1682394017.1483643369