I do have to wonder what you're growing that requires daily watering. But, portulaca won't appreciate it at all.

Which Portulaca are you growing? IME with P. grandiflora, umbraticola, pilosa, amilis, they close up their leaves and quit making new blooms if they get too dry. I had a horrible time keeping up with watering them the past couple summers because it did not rain for extended periods. But what you describe sounds like it could definitely be too much in the other direction. I'm also curious why sprinklers would be running every day.

I live in a condominium complex and the sprinklers that come on to water the communal grass happen to hit my garden as well. I've inquired about moving the jets but that doesn't look like that will be happening.

I figured it was something like that. In that case, if the sprinklers are hitting the whole area, experimenting with a few plants first seems prudent, of either the same type or unrelated ones.

If you were to put a plant in a hanging basket on a shepherd hook, would it still get wet? If not, that could be an option for plants that don't enjoy the daily sprinkle.

Various types of "elephant ear" plants would love these conditions, if your thoughts turn to experimenting with other plants besides Portulaca. If you do try Portulaca, or something else that you worry might not like such moist roots, try planting each rootball a little high, making tiny mounds around each one, so the soil directly around the roots is elevated from allowing water to puddle around them at all.

Trying to reach an alternative agreement or solution with someone who
decided it was necessary to water the grass daily and to place
sprinklers so that water also hits non-mowed areas does sound like a
futile activity.

Happy to suggest. Best luck and it sounds like you're about to make a new garden buddy!