It would be easy enough to run say a GTX328/330 from a laptop-style LIPO battery.
As mm_flynn explains, the transmission duty cycle is miniscule and this also deals with the much advertised health hazard; this isn't a mobile phone which - in a poor signal area - can be radiating as much as 0.4W 100% of the time, and be doing it right up against the side of your head.
Rod1 - how is the empty weight of your homebuilt made up? % due to engine, airframe, avionics, etc. It should be trivial to save 5-10kg by replacing certain non-structural items. I realise you carry a transponder but you gave this example as representing huge numbers of pilots.
On the topic of not understanding homebuilt capabilities, I do know full well that just about anything with an IC engine can reach at least 10,000ft. The question is: how often is this used? The issue in all "mid-air" debates is the statistical likelihood of an encounter, and my experience (~ 900hrs) firmly confirms that large chunks of GA do fly very low. To a large degree, the 0-1000ft AGL band is a fair game (full of various flying lawn-mowers, helicopters, towers, etc, and with ineffective radar service due to terrain) and if you fly there you should accept the greatly reduced safety (albeit still a very low mid-air probability). It is above that which I am not happy with.