Hi and welcome,
Not sure of the connections you would use via Tyros 5. Wireless wise, the most common method is by using a wireless MIDI system. The cheapies really don't work that well, if at all (Midair for instance).. And for the most part are only somewhat useful at a short range. The more expensive (pro) MIDI wireless units do perform quite well.
The next issue is getting the sound (via a sound card) from the computer to where the keyboard is located. Off the cuff, I think you'll find doing this wireless bit is possible, albeit cumbersome and the results might be overall disappointing.
OTOH, my MacPro computer was originally located approx 30 feet away from my console (downstairs) . I used a powered USB cable (actually 2 of them) to convey the MIDI information back and forth. 8 channels of line level audio (from the internal computer sound card) via shielded balanced cable to the power amp located at the console location. It did work, but again, it wasn't the best setup for several reasons. I have since made an installation of everything, computer and sound equipment at the console location. Much improved from an operational stand point. My humble opinion.
Quite possible for the cost of doing the wireless thing, you could probably buy another (dedicated to Hauptwerk and the keyboard) computer . I just bought a (used) Dell desktop with significant high end features on ebay for approx $300+ . This for a friend, helping him setup a Hauptwerk organ. You might have to shop around a bit.
Rgds,
Ed