OP, you seem confused about the legal advice you’ve received from your solicitor. Are they not confirming their advice to you in writing?
If you have a possession order you’ll be able enforce that if he fails to vacate the property by the date ordered. You simply instruct High Court enforcement officers to physically evict him.
As far as the claim on the estate goes, he is entitled to bring a claim under the Inheritance Act if he is either (1) an unmarried partner who lived with your aunt for at least 2 years prior to her death or (2) he was being financially maintained by your aunt (partially or fully) prior to her death. If she had provided rent free accommodation to him, then it is arguable that she was financially maintaining him.
It sounds like he’s issued a claim form. Has he served Particulars of Claim yet? You’ll need to file a Defence to the claim setting out why you say he is not entitled to any part of the estate. If he has no resources of his own to house himself, and your aunt was effectively supporting him, why do you say he is not entitled to reasonable provision from her estate?
Has your solicitor suggested making a Part 36 offer? In your shoes, unless I was absolutely certain that I’d win the case, I’d formally offer him a lump sum settlement of money. There are cost consequences for him if he fails to beat your offer at trial and so it will put him under pressure to agree a settlement. A negotiated settlement also removes the uncertainty of litigation and the ongoing legal fees.