UK registered vehcile will not need carnet.
ALL south american countries you obtain 'temporary import certificate' at borders on entry. Usually no cost. and most countries give you 90 days permit, exceptions being Bolivia (30 days) and Argentina (180).
IDP not required (though handy if you have one perhaps), but generally only driving licence required will be your standard UK licence.
It is unusual to ship into Brazil, obviously possible and too be honest I'm not sure of the reasons why people don't but as I say it is unusual.
Usual ports of entry from europe would be Montevideo (Uruguay), Santiago (Chile), Cartagena (Columbia).
Options for shipping are either container (110 will fit a std container) or RoRo
RoRo charge by the SqM. Grimaldi is the normal and they ship into Uruguay.
I would suggest container is by far your best option.
20ft or 40ft containers are commonly available, obvious advantages are that you can share a 40ft with another vehicle.
Costs vary and will depend greatly on how much you want to do the paperwork or use the services of a shipping agent.
Your estimated cost of 2000 each way for shipping is about right, i'd be surprised if it costs any more than that. It could be possible to ship for half that.
Argentina has just devalued it's currency which makes it a VERY cheap country to visit at the moment.
3 months is not very much time to really see the vast and varied continent of south america. It's BIG over there, don't underestimate the amount of driving and relatively slow driving at that. We spent best part of 3 years exploring there.
Consider buying a vehicle there.
If you only have 3 months then perhaps to save the cost of shipping it could be cost effective to buy another travellers vehicle or even a local vehilce ?
It is certainly possible to buy something for less than say 4000 quid.
hope this helps a bit.