Why wouldn't it work to download the whole blockchain, maybe you can do it in steps, 30GB this month, etc. I honestly think it's not possible for you to restore your coins withouth getting the program to sync with the network, because that's basically what it's doing when it gets the chain.
I use a lightweight laptop without the massive amount of space needed
What do you mean by a paper backup? Do you have your private key? If you do you can use some light client, which does not require you to download and sync the blockchain, and just import your keys to it.
Eg. Electrum is a good choice.
Once you have your private keys, you can transfer your coins anywhere you want to.
I have the Root Key, but I am not certain on how to use it. Do i just import it into Electrum, in which case how?
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