The miner could add just a single high fee transaction and be guaranteed the lottery.
Maybe the network could wait for multiple blocks within a certain timeframe, and select the block that is valid and contains the most transactions of them all. Another option would be that the network only allows blocks to have between 90 and 110 percent of the transactions of the previous block (just like difficulty scales), no matter how long that takes to reach.
What problem are you trying to solve other than "I don't mine so I don't like it that other people get subsidized coins"? The primary purpose of mining is to secure the network. Any system that discourages mining is working against the basic interests of the network.
I'm not trying to come up with a network that discourages mining, that should be rewarded as much as possible. I was just thinking how I could encourage some actual usage too.
It would be nice for people to be able to earn some coins, without expensive hardware, just by participating and some luck. There was a time Bitcoin had lots of faucets for that purpose, but that was just giving away free money to anyone who asked, causing lots of fraud. While my automated approach only gives it to the people that deserve it (those who make actual transactions and paying the fees for that). The end result is that the coins are more evenly distributed across the world population, and you can't possibly be against that?