Does it use less power to process and compress data on a chip, or to transmit it to a station?
It depends on how much power you use to transmit the data, versus how much power you use to process the data.

Given you have such a wide choice of hardware and protocols, and your transmission range and environment is not specified, the question is completely unanswerable, without many more details.

Your way forward is to choose some transmission hardware, and some processing hardware, and estimate their power consumption. Then see whether lower power stuff exists, rinse and repeat. Then buy the best you can afford of each, program it, and choose the best.

A few hints, bluetooth is much lower power than WiFi, though the latter would give you a more capable connection, if you like shh'ing into remote computers. Consider 6LoWPAN as a protocol. Some processors built for smart meters have very low power consumption, some processing, and a radio built in, like a TI CC1110 for instance. For one-off, buying the manufactuer's dev board is a good way to go.