I know from reading on cloud programs, adobe speedgrade can color correct that. Much wana go in that direction. Doesnt look too bad, just a few filters and adjustments and you'll be just fine dude
You might want to consider taking the video into Photoshop to use the new Camera Raw filter. Or, perhaps export to a series of still, adjust one, sync the others, and use the results in Premiere Pro as an image sequence.
But first try the levels effect to see if it recovers any of the blown out area.
Try adding Levels effect and drop the Whites and Mids. (RGBs)
Do not lift the Blacks