I'm probably the guy who was mentioned who commutes from Silver Lake to Santa Monica, cause I do. I go from Santa Monica to West Hollywood and beyond almost everyday.
Lots of people do it via Santa Monica the whole way, but its not the friendliest. If you are carrying a load and not keeping a quick pace, I'd just take an alternate route that I'll detail below.
Here is the route I'd suggest. It is one of the routes I take.
1) When turning from Sepulveda to Santa Monica Blvd, at around mile 4. It is a large intersection with a left turn lane. Depending on my mood, I'll cross over the left(east) side walk when I get closer to Santa Monica Blvd and then just cross via the crosswalk instead of via the left turn lane. Less of a hassle than waiting in the middle of the intersection or behind 10 cars to make a left turn.
2) Right around mile 6 on Santa Monica Blvd in Century City, it is 4 lanes of high speed traffic and you need to make a left across it all. I merge over into the bus only lane and take that. I know some people will cross via a cross walk and then ride the wide sidewalk/path along the left side of the road.
3) Once you hit Beverly Hills all semblance of bike friendly goes out the window. You can continue straight on Santa Monica at this point, which I'd done hundreds of times, but its not fun, or can cut onto Merv Griffen Way right after the bus lane spits you back out onto Santa Monica Blvd and follow the route on the map to Carmelita Ave. This is a nice wide slow residential street that runs of the length of Beverly Hills. Lots of stop signs, but no real worries. (besides an overzealous BHPD officer for rolling stops.)
Once you are into West Hollywood there is a bike lane, its right next to lots of parked cars and backed up traffic so it can get dicey from people not paying attention, but its usually pretty slow traffic due to congestion.
Depending on when you get into town and would be riding that way, I'd be happy to ride along with you if our times matched up.
RE: Weho to Union Station.
I'd take Willoughby Ave to Vine, and then turn left to N Arden Blvd. to 4th St. to New Hampshire to 7th St to Main which takes you to Union Station. Or you can jump on the subway earlier than that and take it there.
Hope this helps. Let me know if you have any other questions on it.