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Possible to have two different checkout processes (and payment processors)?

I've been tasked with creating a checkout process for a country other than the one we normally do business in. This other country has a completely different process for CC payments, so I'm going to have to build a completely custom checkout process for the country.

Is it possible to have a unique checkout process per Magento website? Or do I need to create a whole new Magento installation and make the other country's site there?

I've looked around and couldn't find anything concrete. This Magento answer doesn't apply since what I'm asking is different than the linked question. I've also thought that I could make the other country's checkout in a new module, and have that module enabled or disabled on a per-website basis, but the only place I've seen someone try it was here and the comments mention that turning on caching breaks this hack.

Any insights would be appreciated. Thanks!