The mud chutes probably verge on class 3 - however a class 2 paddler, with instruction on the trip, would likely be able to handle it, but it is a risk. FWIW, it might well be best to swim to the lake with an unloaded boat than attempt to get out, given that it is very, very mucky. Definitely scout prior to the trip.
If the lake keeps rising on recent trends it will not be an issue at all.
There is supposedly an alternative takeout six miles up from the bridge at, as I recall, Sheep Springs. I've been 3 miles up the track, on the west side. I stopped at a spot where I rebuilt the road and decided NOT to test my workmanship.
FWIW, the best craft for this river at normal flows might well be keelless lake canoe - given the high carrying capacity and short draft. I did it with a high rocker whitewater canoe and was able to carry quite a bit of stuff. I walked 75% of the river barefoot on sand (kinda fun, but not fast) - and needed to drag briefly every 50-100 yards or so.