It's not in a good enough state to just hand it over.
Our women's side averaged 3.5k across all fixtures this season. This is obviously slightly boosted by the 8k who attended the United fixture at the Amex but they pulled in over 4k twice this season at Crawley.
The Withdean currently holds 1300 and almost certainly isn't up to the necessary standards required of a WSL ground. It'd cost a lot of money to bring up to that standard and I would presume that the council have no intention of selling the land and ground to the club anyway. Why would the club invest all the money in something they wouldn't own?
A 10k stadium, at Falmer, is probably the only viable option for the club to ensure that both men's and women's side play in the (official) boundaries of the city.