Alternative incentives for miners
Transaction fees, just like in the bitcoin blockchain.

The level of mining would then adjust up and down with the amount of usage.  You might need to subsidize it to get any participation from the start.

If demand is too low where fees don't justify this, then demand is too low where fees don't justify this.

Perhaps you can use Namecoin, which already exists, to solve your problem?  Namecoin can be used for namespaces other than just .bit domains.