Bitcoin in wallet transactions but not on the blockchain (old wallet)
I'm afraid you won't be able to access these funds.

How do I access these BTC and get them onto the blockchain? I could be wrong but I believe that it's such an old unconfirmed transaction that it doesn't even show on the blockchain.

The transaction is not, and has never been, confirmed on chain. From the perspective of the network, it doesn't exist.

So far I have tried forcing the transaction to go through again with a hex code of the transaction on https://blockchain.info/pushtx, but it didn't work

If that doesn't work, it likely means the transaction is spending funds that no longer exist (i.e., it was double spent). The error you get from sendrawtransaction does not mean anything, as your node is not fully synchronized yet.
Regardless, I'd suggest to let your node synchronize with the network. It's possible that it has received other transactions at that time, which did get confirmed.