Does anyone have any suggestions what I can do apart from lose the funds? Is it possible to open a cloud based wallet and somehow redirect the pending transaction? Or is it possible to continue the bitcoin core download but onto an external hard drive?
These are all valid possibilities!
First: Make a backup of your wallet.dat if you haven't done it yet.
1. You can export the private key, and import it into
Electrum for quick access.
DO NOT SHOW YOUR PRIVATE KEY TO ANYONE!
2. As an alternative, you can restart Bitcoin Core with the prune=550 option. It will take only a few GB on your computer, but you still need to complete the 185 GB download (this takes a while).
3. Using an external hard drive is possible too, but slower.
Take your pick, it saves me from typing unnecessary details.
Where did you obtain an address without loading the wallet to begin with?
Bitcoin Core loads a wallet the moment it starts up. It doesn't need to be synchronized to produce receiving addresses.
I'm assuming my current transfer from the exchange to my bitcoin core wallet is floating waiting for my bitcoin core to load then register the transaction.
If all went well, your transaction is confirmed already. Your Bitcoin Core doesn't need to do anything for that, all it does is show you your wallet.
The important thing is that you keep your private keys, which are in your wallet.dat.