1. I don't get the Amazon gift card angle. Is that some allowance that you're giving the employee? Those gift cards can be sent electronically and they're more convenient to use that way since it's a website.
2. You can drastically simplify the pricing. I would remove the two tiers and just make it one price. Increase the price a bit more and have shipping included. Reduce the complexity, reduce the friction.
3. Why stop at new hires only? Market this as a way to keep remote employees happy whether it be at the time of onboarding, just for fun, or whatever milestone/event.
4. Why compare what it costs you vs what it would cost them? The value is more than the cost savings. It's the convenience and the fact that you're made for it. Give the client an option to include a personal note or something to make it go from commodity to premium.