I have a localhost version of every site, even the small ones, and usually use a local Git repo for each one. I lose a little time by having to move stuff to the live server, but I gain way more time than that by being able to use a first-class editor that points out my errors in real time as I edit. It also gives me a solid, unhacked, local copy/backup of the full site.
That said the advantage of using the editor would not be as great if I were not spending most of my time working on PHP code for my extras, especially since I can run and debug the PHP code without leaving the editor.
I think your client would find the dev costs a lot higher if multiple devs worked on their site and overwrote someones code which then had to be completely rewritten. Hitting that commit button at the end of the day at the very least shouldn't skyrocket much in terms in cost, using an agile workflow methodology though will of course be better.
I'd suggest just registering a staging domain, it could even be a subdomain of your agencies primary domain and then start sticking in clients websites within subfolders for their viewing and for your dev team to collaborate with (of course prevent them from being listed on Google and accessible by the public). It really shouldn't cost much unless you're working on multiple massive websites that are going to eat your current servers resources.
You'll then have two backups of the live site, the repo and the dev/staging site so reverting and debugging will save costs too.
