Thanks for your advice, everyone.
Unfortunately I have a limited number of USB ports on both computers (unless I faff with the 'media base') so the rather simpler USB bridge would probably be a bit awkward.
So I'm going to stick with the simpler USB stick expedient. This is because computer A is the primary upon which I work. I don't really need to transfer files between PCs, but I do need to view work created on one on the other - and it's dead simple just to copy the file temporarily onto a USB stick (which lives next to a female socket extension to the other computer) then view it on computer B.
I also need to run a specialised application on computer A and make notes about the results in Word on computer B directly onto the comment document held on the USB stick. When I've finished, I just copy and paste the revised document back to computer A and let it overwrite the original.
This also slows down fat fingers which might lose the original file!