Can a raspberry pi zero run bitcoin core 0.16.1 and sync up without downloading the entire blockchain? I'm looking to run a bitcoin core 0.16.1 node on a raspberry pi zero, but I don't have a 200gb+ microsd. Would I even be able to sync up with the blockchain without downloading the entire thing? Is there a lite version mode of bitcoin core 0.16.1 just for raspberry pi zero? I want to just use it to sweep private keys and make transactions.
You can't avoid downloading the block chain with Bitcoin Core  or any other full node.  That is inherent to their functioning.

You don't have to store it, however, if you enable pruning.  This would reduce your storage needs to 1 GB or less.

If all you want is to make transactions, and you're satisfied with having to trust other servers a bit more, you could use a wallet that uses SPV or gets transaction data from a centralized server.  See https://bitcoin.org/en/choose-your-wallet and look for those listing "simplified validation" or "centralized validation" as a feature.