Isn't this the sort of thing where the club organiser/s should quietly have a word? If its a singaround thing then couldn't they explain that if its someones 'turn' then they should be assumed to be solo unless they state otherwise.
Could it be a sort of general announcement, aimed at the singers, that if they want accompaniment or not then could they please specify before commencing
Or how about suggesting that the evening be devided into two halves, the second being the one where its free for all to join in if the person whose turn it is says so.
Its unfortunate that he reacted badly to your first approach, and that he has not applied the lesson across all the singers. It would be a shame to let the situation go on though, because it could be putting more people off & then the numbers might fall. Also this enthusiastic new player will surely at some point twig to the unspoken ettiquette & be embarassed at having transgressed so often.
The happy ending would be if there is some way of channelling the enthusiasm into encouraging a jamming element when required. As I said at the start - I'd be inclined to expect the organisers to pick up on the problem and find some way round it. Bit of a tough job though...

Kris