Forwarded domains & changing URL links within a site

Hello. I have a client forwarding and masking their domain to a WP site I’ve developed on another host server. The URL window successfully masks my domain name, replacing it with my clients domain on the home index page. However, once a link is clicked the domain name in the URL window reverts back to my domain name. But, I want the domain name to stay the same as my client’s forwarded domain for all links within the site.

So, is there an optimal way (or plugin maybe) that could manage this?

Is there an ideal way to manage domain names (siteurl and home) in order to develop a WP site within one’s own separately hosted domain, while having a site function properly in both ongoing development and live viewing from a client’s forwarding domain name? Again, with all links within the site staying the same as a client’s domain, which would be different from the domain name my site is actually being hosted and developed on?

I’m really at a loss where to begin.

Thanks much,