Is there an FTP or FTP-like client which allows "delta uploads" - i.e. only uploading the parts of a file which have been changed?

Vim has a built-in plugin called netrw, which allows Vim to edit remote files. Rsync is one of the protocols it can use:

:e rsync://user@hostname:port/path
:w rsync://user@hostname:port/path

Of course, if you have SSH access to the remote machine, and Vim is installed on it, it may be simpler to just use Vim remotely. Nano, Emacs, and other editors with command line interfaces are also options in that case. Depending on just how bad your connection is, though, this might not be a pleasant option.

That all said,

making hundreds of tiny changes to files which are remotely hosted and which must be edited "live" and "in place"

without a reliable rollback system such as what git can provide is you and your organization just begging for agony when you make the inevitable mistakes which may very well go unnoticed until you've forgotten exactly what you did.

Beyond Compare is a diff tool that can compare and merge folders and files; FTP servers are supported too and can be handled like a local folder. It is proprietary, a lifetime license for one install costs 30 USD for the standard edition (with FTP) or 60 USD for the professional edition (with SFTP). It runs on Windows, Mac OS and Linux. Personally I develop my website on my home computer and upload the changes to my Web server when I am happy. I go through the changes by hand with Beyond Compare but this can be automated too. You could try it and see if it uploads only complete files or chunks of changes; I am not sure about that.