Looking at the datasheet I just changed the timer from Internal to External :
from this:
' Setup Timer1 for resets after ~65.5mS
T1CON = %00000000'%00000000Internal clock, 1:1 prescale, Timer1 off for now

to this:
' Setup Timer1 for resets after ~65.5mS
T1CON = %00000010'%00000000Internal clock, 1:1 prescale, Timer1 off for now

Now it works and holds the output on while transmitting but when I release the button most of the time there is a delay of several seconds before the output goes off.

Am I on the right track?