After some research and discussion with TI, I can offer some firm and some possible solutions for you.
If you wish to continue using the CC2571, you may be able to use a crystal with a lower wakeup time, most likely crystals with lower ESR and/or capacitive load.
Another possible option would be to ground Q1 via a cap and then connect an external 32.768 kHz clock to Q2 via AC coupling.
For additional hardware support please contact TI for assistance but we have not tested any of these methods ourselves.
If you are able to consider chipsets other than the CC2571, we have tested the nRF24AP2 and it has very fast startup times which would easily have you broadcasting within a couple milliseconds.
The AP2 defaults to the internal RC clock which is very fast compared to an external clock, but testing the external crystal on an AP2 module revealed very fast startup times as well.
By using the external crystal the base current draw may be low enough for your application, and using byte synchronous SPI if possible will also lower your current consumption for the AP2.
If current consumption on the AP2 still would not be acceptable, the AT3 chipset should also give the latency requirement you need at a lower active current than the AP2 but this has not been tested.
Cheers