I’ve been giving this some thought, and wondering what I'd do if I was in the same position - most of which you have probably covered and discarded!
Have you found the birth of Edward’s wife ‘Anne Maria Harris’?
As they were both born ‘out of County’ in 1841, they may have come from the same place, so if you find her, you might find him
On the other hand, they could have come from totally different places, and met in London.
But it’s somewhere to start.
I see there are several ‘Ann Maria Harris’ baptisms around Bath around 1795 - 1805.
I notice on the 1841 Census, she is given as just ‘Maria’, and she is also ‘Maria’ on the burial entry for her and Edward’s son Edward on 30.10 1831, born 7.11.1829.
So not only looking for ‘Ann Maria’ or ‘Ann’, but also ‘Maria’, or even ‘Maria Ann(e).
(And I think on one of the baptisms she is given as ‘Hannah’ too).
Aged ‘40’ on the 1841 Census would make a birth date of around 1797 - 1801, but then again, that may have been an estimate on the part of whoever was giving the information to the enumerator, or she may have been unsure of the year she was born herself, (many people were), so best to give a wide lee-way of dates.
Interestingly. at her marriage in 1827, she signs her name. Quite unusual for this time. So she had had some education. Perhaps from a somewhat higher class then your general agricultural labourer’s family.
Amongst the births is this one;
Maria Harris baptised 17th March 1805 at Bath of Matthias and Maria.
It seems Matthias Harris was an alderman, silversmith and watchmaker, property developer, sheriff of Bath in 1848, and all round big cheese in Bath at this time.
He was born 31.7.1781 at Walcot a suburb of Bath. Father Matthias, mother Anne.
(Some of this info is from a tree on Ancestry. I’m always very sceptical of these, but a lot of the information would be able to be checked).
Matthias (the son), was married 24.9.1798 to Ann Job at Bristol.
Ann died 1800.
He then married a Maria in 1802. She died 1822 in Bath.
Children of Matthais and Maria:
Deborah born 1804 died 1821
Edgar Matthias born 1808, died Jan 1873 in Bath, and Alfred 1820, died 1821.
He then married Mary Higgins in 1837 Walcot Bath, He was a ‘widower; ‘aged 69’ on the marriage entry (he was actually only 56!)
Matthias died in 1874 aged 93.
(Interstingly, the information on Ancestry misses out the birth of Maria in 1805, so they have either not done all the research they should have done, or have not included her because she was not important to their story, or they didn’t know what had happened to her).
It looks as if the Maria ch 1805 was the daughter of his second wife, Maria
It’s also possible he had a daughter Ann with his first wife Ann Job, which is not recorded, She may have died in childbirth.
Matthias Harris, the father of Matthias, died in 1825 ‘of Southgate Street, Bath’. He was born about 1745.
He was of Lyncombe and Widcombe, another suburb of Bath, right next to Walcot as far as I can tell.
He and Anne had several other children all baptised at Walcot.
Joseph 24.3.1776
Etty 15.5.1779
Thomas 26.3.1786
Edmund 8.3.1788
Hannah 4.4.1790
I looked up Edmund on the Census.
He never married
He died 11 May 1871, aged 83, of 13 Queen Square, Bath.
He was a ‘professor of music’!

I wonder if this is how Edward Barnes’ two sons came to be in choirs and composers? It’s an idea.
It looks as if both Matthias in 1874, and Edmund in 1871 left wills.
There are short entries for them in the Probate Calendar, but you would need to send for the wills to see what is in them. Both have executors of a John Stone, (bit difficult to make out the first name on one of them, but I think the same person), who may have been a solicitor at a guess.
Edmund was buried in Bath Abbey no less.
There looks to be obituaries in the Bath Chronicle and Weekly Gazette for them, but I can’t get to access them to see what they say.
How does this help with Edward Barnes?
There is a burial of an Edward Barnes 18.9.1789 at Walcot - the same small place as the Harris family baptised their children , but this is too early for your Edward who was possibly born around 1800 - 1808. Grandfather?
I have not checked if this Edward Banes, or any other Barnes from Walcot, left a will.
Might be worth a shot if you haven’t already done so.