Basically you're saying either the Parks Hwy. or the Glenn Hwy. If it were me I'd play it by ear and go on the Parks if the "mountain was out" (the first really good viewpoints are in the state park, IIRC, which isn't as far as the NP entrance) and you can also go through Talkeetna, or the Glenn with the glacier (which has thinned and retreated quite a bit) if it wasn't. The Glenn leading up to the glacier has better scenery than the Parks leading up to the viewpoints (though the Parks is nice along the ridge heading into Fairbanks later on, especially in fall).

You'll see the state bird a lot in June regardless, but you get the benefit of the very long days. There's also always a possibility the weather could be rainy, in which case I'd do backups like the museum in Anchorage.