You can do APRS for about $50 if you know where to get used equipment for cheap, but it would require a lot of custom work with pretty expensive new parts to build a system that could fit into a bike's seat tube.

A water bottle, on the other hand, could fit an entire APRS setup without too much trouble. Maybe that would be a more viable method.

You basically need a primitive GPS receiver and antenna (no fancy display or mapping crap needed; it just needs to resolve position to lat/lon and spit it out electronically in a standard data format), a radio transmitter to broadcast with (almost any used handheld 2m ham radio works) and a terminal node controller to watch what the GPS is doing and tell the radio when and what to transmit. TNCs are really small and can be assembled for pretty cheap.