These are just my initial thoughts, so I’d be very interested to hear if anybody has similar/different views on bell-ringing privacy.
My initial reaction is that it wouldn’t be inherently contradictory for you to send a Bellboard link to the church. I think for the Clapper reports, which often contain slightly more detailed information than only specifying that a ringer was part of a performance, your practice of just using first names, or name and last initial in my case :-) , is reasonable. I think the difference here is that after performances/quarter-peals, ringers will sign their names in the guest book, which will be posted on Bellboard. I suppose the step that we may need to make explicit is that our performances are uploaded to Bellboard afterwards. I imagine that if a ringer did not wish to have their name publicly listed, we’d be able to list that bell as “Anon.” or “Local ringer” on Bellboard for example.
To address your question, my thought is that because the quarter peal commemorating the royal wedding has been publicly posted to Bellboard, it’s reasonable to pass the link along to the church. Of course, this is subject to any objections from those who were involved in the ringing who may not wish to have this information sent along.