Hi stuarteve, welcome to RootsChat.
The 1813 map was given to me as a present because of my interest in old maps and was obtained from Cassini Maps :
http://www.cassinimaps.co.uk/shop/mapmaker3.asp The map was centred around Littlehampton but went as far north as about two miles north of Crossbush/Arundel.
The map was originally at a scale of 1:63360 or 1 inch to 1 mile and was enlarged to approx. 1:40000.
I couldn't find a reference to Canonbernes or Canon-Berenes but as the map only notes large estates and large farms that's perhaps not unexpected.
I researched Canonberners and found several references to the place and the occupants Eustaces including one referring to the place being associated with the Crossbush barracks. The earliest large map in 1:2500 is 1876 Ordnance Survey but by then the barracks had long gone - demolished in 1832. I also found a probate record for 1933 for a Eustace so it was still there then but 1:2500 maps of 1915 don't show it and the next large scale one is in the 1970s which again doesn't show it.
I note that in the probate the address Crossbush is listed before Lyminster (Leominster on the old maps) and I did wonder if in fact it was there but that would tend to question the connection with the barracks. I had a look on the 1876 map but few properties are named in Lyminster.
Another good old maps site is Old Maps at
https://www.old-maps.co.uk/#/ - you can scroll across the maps which isn't straightforward on Cassini. In the past I've bought A3 size maps from both downloaded in pdf format to then print off via a graphics programme as I don't have an A3 printer. The resolution is excellent and in a graphics programme you can zoom in by several 100% so that might help.
Do you have any information that might help locate it better?