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.
Like others have answered, you don't need to store the entire blockchain, in your bitcoin.conf file, you can just specify the number of blocks to keep and the old blocks will be pruned.
I was doing this last night on my Raspberry Pi 3 b model. I installed Raspbian and then downloaded and installed the btcd full node implementation. It was able to download and store with the full blockchain with no issues. I stopped it before it could fill up my 32 GB sd card. I already ordered an independently powered 1 TB drive that should be able to store the entire bitcoin blockchain for the next 10 years.
I think Raspberry pi zero may struggle with this, but I have a few of those and might give it a shot. I don't see any obvious issues.
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.