i am in avery similar situation. the process that i am going through is as follows. it seems like me you have a bullying ex. like you i own the property in my sole name, paid the 5% deposit, no written or verbal agreement that he would be entitled to 50% of the proceeds. i got a solicitor to write to him and reqest tha he vacate. about 2 months later i received a letter from his solicitor claiming beneficial interest and that they would be registering a restriction on the property and that my ex would provide evidence of financial contibutions to support his claim, which he did and on the disclosure he called evidence his contribution was minimal in relation to the outgoings on the property. we are at a stage now wherebt the restriction has been lifted, and instead signed an undertaking that i would not do anything with the property until all matters were resolved. he went to a mediator, he had to this because of legal aid rules. (you maybe eligible, check it out) over the period i kept detailed financial records of who paid what and when and i suggest that you do the same and then you will get a clear picture of his actual contribution, it will give you a starting point to move forward with it. what you dont say in your post, did yourF ex live in the house at the time of the purchase, or was it later on. from what i understand cohabitation is a very grey area of the law, and cohabitants unlike married couples have to prove their their interest in the property.by virtue of their contributions and their intentions when they bought the property. grt yourself a good solicitor, preferably by recommendation. Or indeed try cab for starters.whatever you do do not offer any money, that is like admitting that you owe him, let him do the work. good luck!!
Iwish you well