Having done this area more than a few times, my suggestion is thus: Why not camp at Hamilton Lake on day 2? Day 1, get as far as Bearpaw: Sure, that is not the greatest camp, but you could always spend some time on the porch of the store taking in the view after the 11 mile hike getting there. It will probably be late by the time you get there, which means most of your time will be spent hiking, making camp, eating, cleaning up and getting ready to turn in. Day 2, take a leisurely stroll to Hamilton Lake. There is plenty to see in this fairly spectacular section,, and after the day 1 slog, this short hike would be a pleasure. That would give you plenty of time to rest, hang out at the lake and maybe take a swim. Hey, the point of the journey is the journey after all.
The next day, get up at the crack of dawn, eat, strike camp and get on the trail. It is 2.8 miles and a accumulated altitude gain of 2451 feet to Kaweah Gap summit. At the very
slow pace of 1 mile an hour, that means in about 3 hours you will be at the highpoint of the days hike. Get out of camp by 7:00 AM and you will be in the gap well before noon. Like at a moderate pace, maybe by 9:00 AM. Whenever you get there, it means plenty of time to get to Moraine Lake and it is mostly gentle up and down, mostly down. So from the Gap it is about 10.1 miles, 1509 feet of gain and 2532 feet of loss to Moraine Lake. Say a ~13 mile hike with the worst of the uphill in the first few hours and lots of daylight. Piece of cake!