Well, Mibew Messenger 3.5.3 with the fix has been released.
Also, I'd be really interested to understand the issue if you're able to explain it to me - I've spent a while really trying to understand why it behaves the way it does.
The problem was in invalid attempt to set cookies for a non-public second level domain (.co.uk). The accurate solution for the issue could have been made using some public suffix list library, but client side cookies creation is made by widget which has no third party dependencies and contains no third party libraries. So it was solved another way. Less elegant, but workable.