Followed some links from your links and found the actual text of the law, which summarizes the law as:
Prohibits person from carrying child under six years of age on bicycle or in bicycle trailer.
Elsewhere in Oregon law, a bicycle is defined as:
[a human-powered vehicle] designed to travel with not more than three wheels in contact with the ground.
So a unicycle, bicycle and tricycle are all covered by the proposed law. There's also a wheel-size provision in the definition, which makes me suspect that too small of wheels or 4 of them puts you in a category like skateboards that probably aren't allowed on the road.
So I think you simply have no solution if that law passes as written. Everything I've ever seen for using a bicycle to transport a child constitutes "on bicycle" or "in trailer". Even if you devised some clever sidecar or push-cart mechanism to get around the letter of the law, I doubt that would work.
But just in case it's useful, there was an earlier question along these lines that I think covers all the answers you could be looking for: Recommendations for Transporting Twins On The Front of a BikeRecommendations for Transporting Twins On The Front of a Bike
Basic summary:
- Coolest option is a bucket bike (long wheelbase with a big box built in). Common in Netherlands, rare in the US. Search for "cargo bike" in your area (I know there's bikes of this type made in Portland, OR)
- Next best is probably a long-tailed bike, such as an extracycle, especially if you can get seats designed for that use
- There's child seats that mount on a bike rack behind you
- child seats that mount on the handlebars in front of you (you could combine this one with any of the other options)
- Trailers are pretty darned good for carrying children.