I like the client-cheating part and this idea could live on its own.

To summarise: "To get a resource from server client needs to play x amount of bitcoin (reusable POW) or spend energy on 2x amount of POW".

This might be a good bridge between old 'unbitcoinized interned' and new 'bitcoinized internet'. Participants are economically incentivized to switch to bitcoin solution, but non-bitcoinized internet also can access the resource.

It worst it protects the server from DOS attack, at best the server earns some money on its services.