You can leave without the bike if it's presence is not noted in your Passport upon entry. Most countries DO NOT attach the bike's TVIP to your passport ... so no one is the wiser if you leave without your bike. Varies country to country however, so check it out.

But even if broken down ... your bike is likely worth a fortune in S. America, an easy sell off to locals who can turn a tidy profit breaking your bike out or repairing it and selling it on. (Some depends what bike you have ???)

But think positive and assume YOU WILL MAKE IT! (most do!)

Shipping is very costly. I'm guessing "about" £1000 ukp each way. Not to mention the major hassle of freeing your bike from port in B.A., then getting it shipped out in Peru'. Time and energy wasted, IMHO ... and most likely will end up costing MORE money than promised. But if you know the ropes and have good Spanish skills ... go for it. (or hire someone to help you!)

If you trade off the shipping cost you could:
1. buy a local bike in Chile or Ecuador, resell legally, fly away. (you can't leave Argentina with a bike bought there unless done between two private parties outside the "system")
2. buy used bike from a fellow traveler. (many for sale on HUBB) re-sell at trips end, fly home.
3. With more time added, buy a bike in USA, (cheap) ride it South, sell off, fly home. (you'd need more than 3 months ... more like 6 months for that option)

With the money saved from not shipping, one of the above scenarios could work. Some riders just can't be without their bike ... but you pay for that luxury. Up to you and your budget.