Lots of Options ... but not enough Time!
If money is tight then no question .... buy the bike in California and ride down from there. This just one option among many. This saves about $1500 shipping right off the top. Bikes are about half the price as in the UK and wider selection of 650 class dual sports and LOTS of cheap bikes laying around unused.
Don't know where you're from ... you didn't say. But if the UK then buying in the US makes sense. If Aus or NZ, same applies.
A rugged, single cylinder dual sport, IMHO, is a decent choice. They will smooth out the crappy roads and handle dirt and mud sections. They are cheap-iss so if lost, stolen or confiscated .... not earth shattering.
Any bike can make it down ... but for best comfort, best fun and most safety, buy as much bike as you can afford. Read here on HUBB and ADV for about a year, you'll soon figure it all out. 3 months not really enough time for all of S. America.
I would start with Mexico and maybe part of Cent. America. Study Spanish,
leave the bike down there and come back again and continue trip with more time and money.
Or ... Fly into Colombia, rent or buy a bike and skip the border hassles of Cent. America and get right into the heart of it. Shipping a bike trans Ocean is a colossal waste of money.