But bitcoin will fork

A soft fork would leave the old signature scheme valid, and quantum resistant signatures optional. People could keep using their old addresses if they want (reusing of addresses is not advised now either, but it happens a lot anyway). The old signatures, in this scenario, would need to be rejected by miners. The idea would be to force everyone to use QR signatures in an attempt to make the largest possible percentage of the circulating supply quantum resistant. Due to lost addresses (estimated to represent 20% of BTC), this percentage will never be 100%.

Having quantum resistance as an option is no option at all, because it would do little to protect the value of BTC. If there were quantum-resistant BTC mixed with non-quantum-resistant BTC, a hack of non-quantum-resistant coins would devalue the quantum-resistant BTC, because they would all be part of the same circulating supply, on the same network. The bottom line will be: BTC got hacked.

The only true option is a hard fork, and the only function of "legacy" BTC would be for owners to prove they own BTC, so that they can claim the quantum-resistant version of their BTC. Again, this would only work if 100% of BTC got claimed / transitioned, and if there was an arbitrary deadline for users to transition their old BTC (which would result in many people missing the deadline and essentially having their money deleted. This group would likely include Satoshi). If not, then a leftover percentage of BTC would remain vulnerable to a quantum hack indefinitely. After such a hack, the attackers could transition their stolen "legacy" BTC and claim QR BTC on the new chain just as any legitimate owner could, resulting in a "quantum-resistant" network where QC attackers might own a significant percentage of the supply, allowing them to manipulate the price severely.