I have a Peugeot307 with heat reflective windscreen. My TT wired receiver failed to work. So I bit the bullet, bought an Arkon powered mount that my PDA docks into and is charged from (so only one power cable) and a SysOnChip Compact Flash GPSIII+ receiver. Even behind the windscreen it gets a very good lock, and in itself is a good reciever with near instant hot starts. Fully recommended, bought from
www.globalpositioningsystems.co.uk Only word or warning is do NOT install the SysOnChip GPS viewer software, you get all the info you need from TT5.
I have had bad experience of bluetooth myself, and as you say it needs powereing/recharging. Plus the CF receiver can be used outside the car should you wish, battery life on my iPaq2210 about 3 hrs.
You could try getting one of the powered mounts that include the GPS receiver as part of the mount itself - but if it doesn't work you're back to a new mount and a new CF receiver.