I would not log in without permission, that could have legal ramifications if anything went wrong. To me it feels like an ethical gray area since they are insinuating a relationship that doesn’t exist, but the legal risk for you not having permission is worrisome. If the edit caused an error, there was a coincidental failure, or if the ex-employer made other accusations later, any of these could look bad with an access log. The risk can be high for you.
Personally, I would escalate the contact attempts to phone and other contacts in the company if you haven’t already. Keeping the tone friendly but insistent. If that fails, and I was concerned about my profile still being on the site, I would have a legal service draft a cease and desist for the ex-employer and their host. Taking legal action may seem extreme, but keeps you insulated from legal risk yourself.