will water softener condensate keep trap filled?

Does a water softener put out enough condensate to keep the trap of a 3" ABS pipe filled (the seal intact)?

yes, the softener drains at least 5 gallons of water every time it goes through the regeneration process (probably every few days)

As I’m sure you know water softners do not have “condensate” water - they do have waste water that results from the regeneration process, this can be a considerable amount of water depending on the unit. Looking at your pictures I would be more interested in the way this is plumbed and especially how the ABS and PVC are connected, mechanically I hope!