Move WordPress Site to a Server Already Running WordPress

I built a WordPress website on my server. It was a redesign of their current WordPress website. I need to move it over to their server now. I also need to keep one portion of their old site which is a user forum. I need the new design and stylesheet of the new forum but the user information, forum topics, etc. of their original WordPress site. What is the best way to move this website over??

Your help is appreciated!

New Site: https://www.acreativepage.com/signal
Old Site: https://www.signalstudios.net