(The refills are less expensive than the whole assembly, and just putting some rocks in the pump basin is easier than installing a whole kit.) If so, how often would I have to change or replace the rocks?
Your question doesn't clarify why you'd rather drop stones in the pump than to replace the stones in a filter cartridge. Regardless, the reaction between acidic water and caustic stone is going to take place anywhere they meet. I've seen limestone dropped into acidic rain water barrels to raise the pH for watering gardens.
The ordinance in my area is that the HVAC condensate must be discharged outside the building, or the condensate must be neutralized before it goes into waste water plumbing. My HVAC condensate is gravity fed to a side yard, with no special stones. The grass there grows like normal so I've never given it a second thought.
Back to your question. If the filter pump is a code requirement, i.e., the water must be treated before it can be discharged into your waste water system, I wouldn't mess with the pump filter system. It's not worth a fine or the liability.
Yehuda_NYC, the cartridges look like the end-caps spin off. They might be designed to be refilled. Putting stones in the cartridges shouldn't cost any more than dropping stones in the pump. The least expensive route is to re-route the distillate tubing to drain outside - if it is in compliance with your local ordinances.