I'm the developer of MobileTest.me, just want to clarify a few things.
As you have correctly noticed, mobiletest.me uses your PC browser's engine to render the website you want to test. This is by design as it was created as a tool to test responsive web design. The mobile devices are provided as an indicator of common smartphone resolutions.
If what you want is to send a different theme to desktop browsers and a different one to mobile devices then you probably need a wordpress plugin such as mobile detector. The problem with this approach is that if the plugin fails to determine the device browser correctly, it will probably send the PC version to a mobile device.
The other approach is responsive design which sends different CSS based on the resolution of a device and does not rely on browser detection. If you decide to go the responsive way you may find this post useful where you can check some examples of how mobiletest.me can be used to check responsive web designs (images are clickable).