One option is, if you're having trouble exporting the private key(s), is to just send all of your btc to another wallet that uses a mnemonic phrase, and allows for keys to be exported. Problem solved.
It works in some cases but it's not what OP is asking.
In case OP needs that specific private key for verification purposes or just don't want to sync Bitcoin Core, he can't do that.
This is why I also asked him on what he want to do with the private key so everyone can provide him a better option,
But sadly, he never replied.