Moving WordPress from subdomain to main domain

Greetings!

I’ve read the WordPress Codex instructions regarding moving WordPress, but none of the information provided seems to apply to my exact situation and I have a couple of questions before I go and break things.

1) I have a main domain and a subdomain. The main domain (www.thomasfielding.com) is a static old (!!) site that does NOT have WordPress installed.

2) The subdomain (wp.thomasfielding.com) has WordPress 4.4.3 installed in /public_html/wordpress/. This is where I’ve done my development of the new site. Users who go to wp.thomasfielding.com DO NOT see a subfolder in the URL.

3) The content for BOTH https://www.thomasfielding.com ( the “old” site) and the content for wp.thomasfielding.com reside on the same server, and they BOTH reside in this user’s /public_html/. Two ‘sites’, one server, one directory tree. Old site in /public_html/, new site in /public_html/wordpress/.

4) I am using cPanel. I am allowed one mysql database, which is being used by the wp subdomain.

I would like to transfer the WordPress site to the main domain and deactivate the subdomain. I want users to land on the main WordPress site’s homepage page when they go to https://www.thomasfielding.com WITHOUT the URL reading as https://www.thomasfielding.com/wordpress/.

Is there a quick way to accomplish what I want to do, or do I need to back up the entire wp subdomain WordPress installation (which was going to be my first step anyway), and then treat the www main domain as a “fresh” install?

Oh, one other thing. My host’s server has a strict timeout limit that makes the Duplicator plugin fail, so I have to manually download and upload all my site content via FTP. The client is a composer, sells his sheet music, and needs large, high-resolution images that cause a server timeout when trying to use Duplicator.

I *can* do all this the long, hard way and have before (site backup, FTP content, install on a localhost server running XAMPP, upload the package, install WordPress on https://www.thomasfielding.com, run the Duplicator installer). I’m just looking for a way to minimize bandwidth usage during all this.

Again, I’ve read the Codex instructions on moving WordPress and the processes described there don’t seem to address this specific situation. That, or I’m misreading something. In any case, I’m not quite certain exactly how to proceed after I back up the “new” site and I’m in need of some advice. It’s my understanding that there’s an “easy” way to do this if cPanel is installed (it is) but I can’t seem to find the page again that said that.
Thanks!