Well, of course it is possible that some wires got crossed ... but there are references to Bella and opera in The Era. For example, an advert in The Era for 16 April 1876 says "Bella Tremaine, Singing Walking Ladies, available for Burlesque or Opera Bouffe".
To be honest, I'm not sure what "burlesque or opera bouffe" is!
So, she sang, she was associated in some way with opera, but there is no record that she sang for the D'Oyly Carte, and she wasn't Annie Tremaine, who was much more famous, and who did!
So I think the assocaition is a family legend, but whether it arose simply from an inflation of a mother/grandmother's career, or from a mix-up with someone with a similar stage name, we can probably never know!
Cheers
Tim