I understand that there is a Restraint of Trade clause.. however, the general nature of these clauses is that they restrain the vendor from opening up a competing business, or participating in a competitor business within the restraint area for the restraint period.
From what you have described, the Vendor is not competing against the business.
In my view, the cause of action you might have relates to a breach of confidentiality clause within the Contract, and/or the lease that would operate to prevent the vendor/landlord from sharing information regarding the business.
I would need to see the Sale of Business Contract, and the Lease to advise further.
Outside of this, it could be a case of defamation. However, it is a defence to defamation if it is the truth, or a "fair comment".. hence the above breaches, in my opinion would be your best way forward.
As such, your next step, once we are confident that there are breaches, is to send the LL a written warning letter.. regarding their actions, requesting they cease and desist immediately from repeating this conduct. It would state that you reserve your legal rights with respect to the breaches.
Please let me know if you want to investigate the potential claim for breach of confidentiality further, via my provision of legal services - starting with a review of the relevant docs, and feedback via a phone call. I will send you an offer of additional service...
Kind regards, James