More than just trying to decide which site to show to mobile users and how to redirect them between the sites, the question you should be asking is what is the content users are looking for and how best can we convey that on our mobile site so that users can get a comparable experience on both ends. There are also multiple problems associated with redirecting users from a Mobile site to a desktop site unless your desktop site is well structured. A good example is given in this article :
"If you're only delivering a subset of content and features on your
mobile site, redirecting users to the mobile URL can cause huge
problems. This scenario happens all the time: the user searches for
something on Google from her phone, and finds the content she's
looking for. Tapping the link in search results takes her to the
mobile site, but because that content doesn't exist on mobile, she
gets dumped on the mobile homepage. Sure, she can link to the full
desktop site, but she winds up on the homepage there too. She knows
the content exists, she just can't get to it anymore. Redirecting to
the mobile URL breaks Google search. How is that ever a good user
experience?"
In this context I also recommend looking at this article A User-Centered Approach To Web Design For Mobile Devices
That said, I strongly recommend against going for the popup route as there is a lot of evidence that popups in mobile are no better than popups in a website. To quote this article
I don’t know about you, but nothing makes me more likely to close down
a website on my mobile browser than an errant popup image. Although
these graphics can be useful on traditional websites in terms of
capturing new subscribers and driving conversions, they’re an absolute
nightmare in a mobile environment.
As you’re developing your mobile website version, make sure all popups
are turned off. Heck, even if – after reading this article – you
decide not to take advantage of all the benefits mobile website
versions have to offer, consider turning off popups on your main site
(or using a tool that deploys them on traditional browsers only) in
order to avoid irritating your mobile visitors.
A better option would be to provide a link which would allow users to direct to the mobile version of the site. You also need to ensure that users can redirect back to the desktop site in case they are not able to find the content they are looking for. I do like the concept of remembering the user requirement but you might find it a tad tricky to implement if you go with a link based approach