Joomla provides a nice face to its interface for inexperienced users, but the backend is very messy depending on the changes you need to make. DW is not going to do what you are looking to do. It's main purpose of Live View is just to allow you to view linked files, and preview the content using the Live View rendering engine where prior, all edits were made in code not being visible until you updated them. Personally, I use the Search function in DW to search the folder to find the right file, but if you only download a particular file to work on you won't have an easy time going about doing this. The best way for them to implement this feature would be to cache the files better, kind of like how Spotlight works for Mac. But that would require a lot of scanning which would be bandwidth consuming and resource consuming for remote file work. I could see them implementing this on a local site where all the files of the site are on your computer though. Based on how Adobe plans to utilize the Cloud, with constant updates, I'd recommend you submit a feature request for additional coding support. https://www.adobe.com/cfusion/mmform/index.cfm?name=wishform
Can I ask what types of changes you are making for these clients? Are you building/editing modules, or working more with templates in Joomla?
99% of the time I am hunting down files in Components, Modules and plugins. All I know at the start is what the screen looks like, and what extensions are installed. Usually thats enough. But occassionally, I just cannot find the code - usually because it is placed somewhere I do not expect it to be. When it gets really tricky is when the template overide system is not used and the extensions have been modified. Template changes are relatively rare (other than, of course, css changes).
The process you describe is pretty much I currently do. The way I like to work is to to NOT download the entire site. Then Dreamweaver creates a local file only for files I have opened. This is really handy when you go back later and you have a repository of only files that have been previously opened. It helps you remember what you had done last time and where the files are.
But when I just cannot find a the source, I will download an entire section and then do a local search. It works, but I was hoping for a time saver...