If you are planning on spending more time around Badlands/Rapid City you could visit them on the way and come back via the Tetons. Although unpredictable, generally speaking the nearer to June you arrive in Yellowstone, then more of the park will be open for 'summer activities'.

The NP Rangers are kept well up to date with forecasts and any risk of a late winter storm, so whereas it is possible you could face some disruption, it is much less likely you would be 'stranded' for days on end in Yellowstone, if at all, as long as you keep updated.

If you have any other questions about your itinerary etc, just ask.