You cannot directly view the Slingplayer, but if you have a computer and wireless connection you could play the video on your computer and have it display through the Wii using a service like PlayOn (www.playon.tv). Another option would be to connect your smartphone directly to your TV via the appropriate cable (I know one exists for the iPhone, I imagine other smartphones have similar options) and then use SlingPlayer Mobile.
That said, bear in mind that the TV feeding the SlingPlayer will have to be on the same channel as the device viewing it, so you couldn't have upstairs and downstairs watching different channels at the same time. If that is something that would be an issue, you're probably better off getting another receiver for upstairs and looking for other apps that let you stream TV shows. (And I completely agree with you about the cost of the SlingPlayer Mobile app - when the slingbox itself is so expensive, having to pay another $30 to be able to watch it on your phone is a little painful. BUT the mobile app is pretty awesome, and with the Slingbox you would be able to watch all your channels when you are away from home, so.. it's really a matter of weighing the pros and cons.)
Personally I am vexed that Netflix manages to provide streaming on the Wii (for FREE with your account) but Slingbox is not available on any gaming devices yet.