The most straightforward way
 

Am I being naive in considering airfreight my elderly UK registered non standard and fully legal BMW R80 into BA, getting some sort of 3rd party insurance, rumbling around Argentina & Chile & shipping the thing back is the simplest & least paperwork way of SA travels?

That is a fairly straightforward way but will cost you quite a bit for the shipping. The beauty of it is that on your own bike you can cross borders, go to Chile, go up to Bolivia, wherever. On a rental bike I'm not sure you can cross borders as freely?

Also, rental rates, IMO, are too high, at around $100 a day, give or take.
This is absurd for the bikes they offer. I can rent a $20,000 USD Harley or new BMW GS for that amount here in the US.

This is pure greed.

You're scenario working out also depends on your time schedule. How long can you spend in S. America? If you are going for a couple weeks or a month, I would think about renting. But if you have more time then bring your own bike over or buy a bike there.

If you're on a budget then by far the least expensive way is to fly into the USA, buy a bike, and ride south from there. This is a long trip but works for some.

Another option is to buy a bike from a fellow traveler when you arrive in S.
America. This has some complications and unknowns but is possible and has been done many times.

I would not become attached to your R80. Lots of other bikes would end up being better long range touring bikes for S. America. Right now there are a couple KLR 650's for sale here on HUBB. Good travels.