I don't think you're understanding how cryptography works. A commitment is basically a hash that is both binding and hiding. In this example it's probably easiest to think of it as a hash. So you hash your post-quantum public key (something like falcon-512) and then sign that hash with your actual bitcoin private key (ecdsa, discrete-log, not quantum safe) and then publish that message to the bitcoin network. Then quantum happens at some point and bitcoin needs to migrate but where do funds go? Well you reveal the post-quantum public key and then you can prove that funds from the ecdsa key should go there. From a technical perspective, this is a complete and fool proof system. DoSing isn't really a concern if you publish to the actual bitcoin network and it's impossible for someone to use up the key space (2^108 combinations at least).

The reason this is a dumb idea is because coordination and timing. When does the cutover happen? Who decides which transactions no longer count as they were "broken" b/c of quantum computing? The idea is broken but not from technical fundamentals.

Yes, that would be a concern. You could require a proof of work to submit a precommitment, so that DoSing was at least expensive to do. You could have some sort of deposit mechanism, where a precommitment would lock down 0.1 bitcoins (from a quantum-secure wallet) until the precommitment was used. I admit I'm glad I don't have to figure out those details.