I don't think that jQuery Mobile will work with "invisible redirection".
Please understand just what "invisible redirection" is. It's a horrible hack, IMO, and shouldn't be used. "invisible redirection" isn't really redirection. It is impossible to "redirect" to a URL, as the DNS system knows nothing about URLs - only IP addresses and domain names.
Your web hosting company or DNS hosting company or registrar (whoever you are setting this up with) has a special web server for this. When users access your site, they aren't accessing your site. They are accessing this special webserver.
You set your DNS A record for whatever domain you want the user to see to that webserver.
That server serves a page that "frames" your site in an iFrame. The user is getting a page from the provder's server that has an iFrame, and that iFrame is loaded from your real site. Thus, the real URL is hidden from the user.
This sort of "redirection" is typically used when a site owner wants to have a domain served from a subset of URLs (i.e. not from /) from a different domain, or when using free hosting, redirecting to a shared blogging platform where the platform doesn't offer domains, etc.
I don't understand why you want to do this. I wouldn't recommend it. In addition to delaying responses, I think it has a negative connotation to users, as this is often associated with "shady" sites.
If your site is example.com, users will not object if on a mobile device they are redirected to, say, m.example.com and see that in the address bar. In fact, it's an expected behavior.
Set up your server to handle a proper separate domain for mobile.
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