Dear Doreen
Apologies for being thirteen years late to respond to your request, but I think I might be able to help you with your request, if you haven't already found what you are looking for.
I have recently been conducting some research into my own family, Baré, and have hit a connection with the Halls from Colchester in the 1790s. My 5 x grandfather William Baré married an Ellizabeth Halls (parents Robert Halls - Mary Nash). Halls has been a prominent second name in my family ever since.
Their first son, Robert William Halls Baré (who was baptised Bear as you point out, no doubt due to the Napoleonic Wars) and his family moved to London where they continued to be fishmongers in the West End. Two generations later, they moved to New Zealand.
Though as pointed out above, many Barés remained in Colchester, with Joseph having the biggest family. There are also a number of Baré families in London who remained.
I would be very pleased to share information with you and in kind to learn from you.
Kind regards, Nicholas