Thankfully I'm working downtown now, but a few years ago I was doing a similar commute. On the evening westbound commute, I pretty quickly learned to avoid that Evergreen Point bus stop on the way home because as others have said you could end up waiting quite a while before you get your turn on a bus with an open bike slot. So, I just started riding to the Kirkland Park and Ride at 108th Ave NE and NE 38th Pl. It's a couple stops before the bridge, so the busses weren't as full when they got there. I rarely had a problem getting on the first bus that came by. There may be another bus stop that's more convenient for you, but just find yourself a bus stop that is a few stops away from Evergreen Point.
As for the morning eastbound commute, when I was doing it it wasn't too difficult to get on a bus at Montlake. Sometimes there would be a couple of bikes ahead of me, but generally it didn't take too long to get on a bus.
I also used to ride around the north end of the lake once a week or so. It would make for a pretty long commute five days a week, but if you just do it once in a while it's not bad.