The new roundabouts built in my area are configured for bicycles to exit and use the cross walks for side streets. They then usually "drop' back into a bicycle lane.
This example shows the bike lane exiting onto the sidewalk.
Here is a street view. That fuzzy white blob on the street light on the right side is a sign that reads "Bicycle lane ends."
It's not official, but it looks to me like the round-a-bout is designed to keep traffic under ~20mph.
If I am traveling above 12mph, I ignore the exit signs and take the lane in the round-a-bout until I take my exit and then move back into the bicycle lane when it becomes available. If I am slower than that, I take the sidewalk as indicated by the signs.

The signs read 'bike lane ends' and 'bicycles use sidewalk'.

Am I courting the flames of danger and riding like an outlaw?

The most important question... Can I get a ticket for this?

Ask in the Advocacy and Safety forum. Not a clue here. Most roadies will probably just muddle through in the traffic lane.

I'd check your Washington statutes and local ordinances.

Check the local statutes. FWIW, if it were me, I'd just take the traffic lane.

Thanks. I'll post over there.
Please drop thread.

I actually sat in a round-about (traffic cirlce) design course. Without knowing rules of your particularly state I can tell you that the design is the typical design called for by high design manual. They asked the class what people would do who came across it, I said merge with traffic and then go through as a vehicle and another person said go to the side walk, both are acceptable methods of navigating the intersection. The are also designed to keep vehicles around 20 mph as noted in general so both are relatively safe, plus no stopping and all legal. Do watch for cars entering onto your right-away though. Been cut off plenty of times in the good old rotaries both in cars and on a bike. Oh the joys of New England.

+1 to checking local laws, but if it were me, I'd check the sign for a code reference as well. Some cite the ordinance, some are just suggestions.

Check the laws. I have two roundabouts on one of my favorite rides. The sign posted says bike lane ends. That's all. So as I approach the roundabout, I check behind me for cars. When it appears safe enough, I take to the middle of the lane and proceed through the roundabout. Once clear I pull back into the bike lane when it picks back up.

I take to the middle of the lane because I have had people actually try and get around me in the roundabout. Aholes for certain, not remotely thinking of the danger they are creating. I'll bet there is no law stating you cannot maintain your ground on the road. Be careful either way.

Bike lanes are more hassle than they are worth. You always have to watch for them to suddenly end and then merge back into traffic. I prefer riding in normal traffic lanes. But then, I also live in a moderately rural/suburban area and I stay off the busiest roads, I'd probably feel differently if I were riding in a city.

I would certainly never use the sidewalk, though, that's just ridiculous. A bike is a vehicle, and you should use the roundabout as if you were driving through it.

To the OP, it is fine for a cyclist in Washington to take the lane in a traffic circle. I do it all the time. You don't have to use the bike lane at all if you can ride at or near the pace of traffic, which is easy to do in a traffic circle. You are generally obligated to stay as far right as you safely can if you can't keep pace with
traffic, but even then a traffic circle is a prime example of a place where it wouldn't be safe to stay to the right.

I'd take the traffic lane.

You can't get a ticket for this unless there's a law against it, and that law is enforced. They can't be bothered to enforce red lights here. Hell, Seattle's police won't even enforce some of the drug laws. I don't think you have much to worry about from a handful of seconds in a round-about. Especially since bikes can get through them more quickly than cars, who invariably slow to about 3 mph because they're so much wider than us.

I would call the city planning dept and/or police and ask. It's not a big town, they are probably approachable.

The roundabouts in Port Townsend are just like that too. Very annoying. I just blow through them in the automobile lanes. I go faster than the cars through there anyway.

Take the lane. No way am I going to exit from the flow of traffic, then suddenly merge again onto the street a few seconds later. That's an invite to get flattened.

If you need any reason why this is good idea, take a look around in that second googlemaps pic. You'll see a truck pulling a trailer and driving in the bike lane. That kind of quality driving is why you need to ride defensively. Watch your ass out there!

Unless there is a sign that specifically forbids it you should be good. Sidewalks aren't safe for cyclists.

To be clear -- I was summarizing the relevant provision in the revised code of Washington in my post above. Unless there is a local ordinance in your town prohibiting you from using the road (which is very unlikely), then you have the right to ride in the traffic lane (generally staying as far to the right as reasonably safe) and to take the lane completely either when needed for safety reasons or when you can keep pace with traffic.

Here is a link to the relevant section of the RCW: https://apps.leg.wa.gov/RCW/default.aspx?cite=46.61.770

And this is also relevant: https://apps.leg.wa.gov/RCW/default.aspx?cite=46.61.755