I just want to chime in and backup runner, I don't have the quite the experience that he has, but I have mowed quite a few lawns, and have only damaged one head, that was stuck in the up position. I run over them all the time without issue.
I use a Z everywhere I can, because its the most efficient way to mow lawns. In any business, efficiency is important, as long as its not causing damage. I would check with your sprinkler company about any warranty you may have.
BTW, I do have 4 customers who insist on a push mower, and I am happy be flexible and do it for them, but their rate reflects the extra time it take to mow with the push mower. They all know this and think its fair.
If you ask your contractor to use a smaller, less efficient machine, expect to pay quite a premium for it.