Hi Josh,
You are correct about the issue with buying a bike in Argentina, contact Sandra at Dakar Motos in BA for details and confirmation.
Yes, bikes bought in Chile can cross the borders easily. *BUT* make sure you follow the process to get it in your name as you cannot enter Peru unless the named owner of the bike is with you.
The process is well documented here.
Guide to Buying a new motorcycle (or a car) in Chile for foreigners | Ondrej Jurik
Our own tale(s) (and where we went wrong) are here
Horizons Unlimited Motorcycle Travellers' Stories - Two Pegs to Patagonia in South America Revisited (Pegless to Peru)
So, a lot depends on how much time you have.
I can put you in touch with a mechanic in Santiago who will help you buy one, he is very trustworthy.
If you get the RUT and start the ownership process immediately, you can arrange to collect the bike paper work in Northern Chile at the offices in Arica (or have them posted to a Hostel, Sunny Days would probably do it).
That gives you approx 2 months to piss around in Chile, Argentina and Bolivia. *BUT* make sure you visit a Notary with the previous owner and get an authorised letter saying you can cross to Argentina and Bolivia with the bike.
Or, if time is an issue, weigh up the cost of taking your own bike over. When we looked into it in 2012 it would have cost about £3800 to fly a bike (return) to the UK. So spending £3500 on a bike in Chile made sense.
HTH