I hiked that route last August and recall a plentiful amount of water avaialble. I had a 70 oz Camelbak and didn't need to refill until I was down from the switchbacks (37, by the way - I counted). I refilled at a stream crossing about halfway between the switchbacks and the trailhead on the return. I think I also recall a stream to one side of the switchbacks. Even without the streams, with a decent size bladder you can probably cover the 5 miles to the lake with plenty of room to spare. With all the snowmelt going on, you should be just fine.
Enjoy the view from the pass - it's majestic.