I tried some unbranded carbon handlebars, but they are super stiff. Did eventually find some rubber filled metal bars for comfort on my favorite ride:
https://a.co/d/0fBY217

For carbon, I suppose it depends on the design. They can be made vertically stiff or vertically compliant depending. The ones I got were super light, so I put them on a commuter bike I have to lift up on to a bus bike rack every day.