Hi there
Sorry to disappoint but even if your ancestor wasn't buried in a public grave, you certainly wont find much in St Mary's churchyard now. It has all been grassed over and all the headstones removed around the perimeter. Very, very sad.
Some of my ancestors were also bargemen in Ware. Ware had a thriving malting industry (not that you would know that now) and the River Lea gave access to London. I also made an enquiry of Ware Museum some time back but their reply was in the negative which I found completely surprising in view of the fact that there were so many bargemen and maltsters in Ware in the 1800s.
Did also try to do a bit of research to establish who actually owned the barges that went up and down the Lea i.e. were there barge owners or were the barges owned by the malsters themselves. Never really drew any firm conclusions.
Valda may be right about the "of independent means" indicating that your ancestor didn't want the enumerator to know her circumstances as from what I know a lot of these bargemen lived in the very poorest areas of the town. Having said that though, Crib St does back onto the River Lea so perhaps your chap did actually own his own barge. On the other hand though, Kibes Lane is the next street along and it is described as one of the most poor and densely occupied areas of the town.
Sorry this is all a bit in the negative but I know only too well that the staff at HALS will be most helpful and you certainly should be able to find something in the parish records. Also enquire about the Poor Law records for Ware. I haven't actually looked at them and know some parish poor law records are more extensive than others, but if he was ill and unable to work, then he might have had to apply for parish relief. Just a thought.
Benz