@ericvdb said:
The problem is that when powered with 12V, the transmitter does not see the 5V on the data pin as high.
Then maybe you can try on 9v as @TimO has suggested, this may be close enough to register the highs and still give you the range boost needed.
Otherwise it would seem you will need some form of line level shift circuit as @dbemowsk has shown.
The only other option would be a 5v transmitter like the Dorji or similar