How can I protect my funds in the event a quantum computer breaks the ECDLP that Bitcoin relies on?
We're deep in speculative territory here but, theoretically, if the elliptic curve algorithm were broken the entire Bitcoin system would immediately collapse.

Let's consider the case where you've successfully guarded your own funds somehow. How many public keys would still be out there vulnerable to such an attack? 

Even though your "base" survived the attack, every other user out there who had UTXO on reused addresses would've been nuked and Bitcoin would have collapsed. You'd be the exceptional survivor. It'd be all over the news, dollar value would've gone to near zero, miners would stop and cut losses and so on. 

So, in my view, there is no way to protect yourself individually against the eventual breakage of the elliptic curve discrete logarithm problem. Even if 90% of Bitcoin users had unspent UTXO's in safe unused addresses, the 10% that were hacked would cause immense damage to Bitcoin in a social sense. 

Since Bitcoin obeys Metcalfe's law, network value being number of users squared, if enough users dropped out of the ecosystem, due to this kind of development in cryptanalysis, it'd surely deal a major blow to the community. 

Not to mention another possibility: such a quantum computer could try brute forcing astronomic quantities of random numbers into private keys all day. Eventually they'd stumble on valuable wallets which could cause the community upheaval of "having broken Bitcoin" and its consequences. So having a known public key wouldn't necessarily be the only attack vector. I'm sure there are other possible attacks using quantum computing, perhaps unknown unknowns that would add immense risk to the system.

Math is the Bitcoin guardian, if it's somehow found to be vulnerable no man would be left standing in my opinion.