I know nothing about Joomla, so I cannot give specifics, but as no one else has given any advice, I'll offer some general comments. Firstly, it may seem an unhelpful comment, but why on Earth have you got an incomplete site in public_html in the first place???? If you must develop on line (and I presume you have to) use a development area, e.g. public_html/wip/, You have to change the structure of your site, so I'd urge you to treat it as an opportunity to put in place a suitable long term structure. Writing amendments directly to the live area is silly, and just asking for a disaster.
Note: I would imagine that Joomla supports that strategy, but if it doesn't, don't use it!!!
So I'd suggest the first thing to do is to set up a "wip" (work-in-progress) area and copy BOTH sites into separate sub-folders of it. Then check that both sites work in the development locations. Then find out how to publish new and amended sites to the live area. If Joomla does not have facilities to do that, perhaps your web host has?
Lastly, don't forget to put a "disallow" for the "wip" area in Robots.txt to tell web search bots not to index the development area. I hope this helps...