I wouldn't make too much of a fuss over it, if the lawn REALLY could wait another week. like during a drought, im not going to stress a drought stressed lawn by mowing it.
but if it does need it and they still want to wait another week, tell them it will be extra because of the extra time it takes to mow it. I had a customer who couldn't afford it right then, so I told them I could put them on my mailing list and mail the bill once a month, but they didnt want to do that, so I told them it would be extra if I skipped that week. long story short, they dropped me because of that. pissed me off, because it was so STUPID! people are really hardheaded! that's one of the downsides of doing residentials.
My set up:
-2007 Chevy 2500hd 8.1l 4x4
-7x18 enclosed trailer
-Grasshopper front mount 61", gravely 52"
-36" Gravely Walkbehind, 21" exmark push mower