You can rent at a local shop. No sense buying at this age especially as she has already "protested." I would make sure she's got snow pants (mamajohns), a proper coat, some thermals, a sweater, and a neck gator.
You might consider the nursery or all day lesson. The ski school is sensitive to the needs of little ones. They schedule frequent bathroom breaks, cocoa breaks, lunch, naps, etc. the total amount of time on skis/ snow is not so long as to be challenging.
think back to your first time on skis. getting in and out of the boots alone is enough to make an adult want to melt down. the all day lesson will be largely about playing and balancing and getting acclimated. its a hoot.
children are extremely adaptable. Unless you or a member of your group has experience in teaching little ones how to ski, i think the ski school is the preferable and safer option. (i've seen well meaning adults fall on kids in tethers). have confidence that despite the fact that she is not a day care baby, she will manage the lesson just fine. as with many other things, it will probably be harder on you than it is on her.
good luck and have a great time.