The LAST_RUN is typically set on a channel that has a script as an input. Most of those channels are looking for records in a DB from the last run until this run.
For your implementation the simplest way to get what you want is to add a function in the channel to set a Last Message time stamp here are a few options.
1. Setup a standard mapping function

2. Setup a custom mapping function in one of your scripts here is the code:
channelCache.setValue("Last Message", qie.getSystemDate());
3. Add it to the ack script on the destination if you want it to verify message was sent first using the same code:
channelCache.setValue("Last Message", qie.getSystemDate());