Is it possible to get bbciplayer working on an arm v6 device?
Highly doubt it can be done, since Adobe Air v3 targets ARMv7 instruction set, its like comparing chalk and cheese in regards to both ARMv7 and ARMv6 instruction set. 

So the chances of back-porting to the older instruction set is difficult to say the least, not alone that, it is a proprietary software so no going in there.

The trend is there and is to stay this way with the advent of increased hardware, supply/demand which dictates support ARMv7 and let ARMv6 die off.