An automotive MAP sensor probably would do, for the data.
But they tend to want +5V. So that's two lithium coins plus
a LDO, still probably OK.
When you say transmit, and you want low power, the thing
that probably has to give is the range (or packet rate). Do
you know how fat and how often the data must be sent?
Because I think I've seen bluetooth microcontroller cards,
Arduino or something else, that might come close to the
weight (and be able to take the 0 - 4V pressure signal and
do the arithmetic) but Bluetooth is not for long hauls.
Now, maybe you could do better by something like using
the 0 - 4V signal as a FM modulation input, to an unlicensed
band of convenience, and get more range & minimal weight
in the electronics. Do the processing on the ground, so long
as you can keep calibration (might want to add some sort
of periodic calibration states, use an analog mux to feed
the FM transmitter 0V, 5V, real MAP V, and then you can
get some interpolation endpoints continuously? But then
you have at least a mux and a clock added to the weight.