I don't believe your proposal reduces energy consumption. Energy consumption in PoW is limited by the value of the block reward
This is a good point, but slightly incomplete: it's the total of resource consumption that depends on the value of the block reward. So if the energy consumption goes down, competing miners would buy more hardware, which moves the pollution to a different location.
The electricity will be reduced by the fact that Round 1 is not computing-intensive and most of the miners won't be "allowed" by the rules (if followed) to advance to the second round.
What's stopping the remaining miners from doing a "51% attack", trying to beat the select few by creating their own new Round 1 followed by their own Round 2?
We can imagine that a lot of fastest miners will always be in the second Round 2. So those will win the race for the Block. In short the honest and fastest will win in the long run.
Those disqualified in the first Round will lag behind and the block they will create will be a fork and it will discounted by the network as most miners will always go with the longest chain.