The trick is to disconnect the aerial labelled BT from the bluetooth module and connect the spare WiFi aerial instead.
I had my 2006 Mac Pro replaced to try and fix this issue, though the replacement had the same problem, and an authorised repair centre couldn't fix it either. So I switched the aerial myself.
I've been happily using a Bluetooth mouse, keyboard and graphics tablet with mine for the last few years. The current aluminium Bluetooth keyboard and Magic mouse work fine.