Some ideas been going through my head.
Have you found out about how Edward's two sons managed to become choristers and composers of music?
It doesn't sound like the normal thing you'd find in a family in London in the mid 1800s.
Normally sons would be off getting a job to help with the family's expences.
How did a wheelwright afford to do this? - Loss of income from them, PLUS fees presumably.
It must have cost quite a lot to go and train to be an organist and professor of music.
How did Frank come to be a 'chorister at Chaple Royal' (from 1851 Census)?
And by 1861 , Edwin is an 'organist and Professor of Music.'
Surely you couldn't just walk in and say 'I want to join the choir', or 'I want to play the organ'!
I can see how Edwin's son could then follow in his father's footsteps. Once you've got your foot in the door...... (sad story about him...)
Have you looked for some history of musical people in the family before this?
How do you get to be a 'chorister of King's College'? Do they have any records?
You never know what information might be on them.
Bios from:
https://books.google.co.nz/books?id=WSLUAwAAQBAJ&pg=PA21&lpg=PA21&dq=%22Edwin+Barnes%22+organist&source=bl&ots=RTrVLWNlYq&sig=ACfU3U3SDah8caFZfwTOzSbnpyhCDdPkHA&hl=en&sa=X&ved=2ahUKEwi-183K-PPiAhUCfSsKHcaiC9MQ6AEwBHoECAgQAQ#v=onepage&q&f=falseAnd:
http://composers-classical-music.com/b/BarnesEdwin.htmhttp://composers-classical-music.com/b/BarnesFrederickEdwin.htmHave you got any of the music they wrote?
Have you found an obituary in the papers for Edwin? Or even Frank or Edward?
Surely there is bound to be something about Edwin's son?
And did Edward , or Edwin etc leave a will?
Again, you never know what information might be in them.
I see James was a 'seaman' (lovingly transcribed as a 'slaman' on findmypast!!).
Have you found any papers for him? Royal Navy? Merchant Navy?
Again, you never know what might be on them.
Good luck!