Haven't looked at the code, but I'm pretty sure there's also at least a basic wget binary in the base Kindle system, wouldn't that be sufficient?
EDIT: Okay, looked at the code, and, yeah, no, pretty sure busybox's wget doesn't handle cookies either
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So, yep, building a curl binary shouldn't be much trouble, there's no real dependencies to speak of if you don't need SSL support. (And if you do, it supports both gnutls and openssl, and while openssl is a bit of a nightmare to cross-compile, gnutls, being a GNU project, should be using a sane buildsystem ^^).
EDITē: Here you go (targeted at a K4/K5, it won't run on older devices). (OpenSSL static linked to avoid the OpenSSl 0.9.8/1.0.x mess, which explains the size). [I didn't test it, but it looks okay].