Cleansers that foam do so because they contain detergents, and this can be a problem for sensitive skin. I also have sensitive skin and I've learned to stay with mild cleansers. I agree with MM that you should stay with the Laura Mercier cleanser because it works for your skin.. There are other non-detergent cleansers available, so you may want to take a look at Dr. B's book, or you can go to Beautypedia to search for non-foaming cleansers.

Thank you so much, Kay, MM and Georgia. This has really been helpful and I'm so glad I posted.
It certainly could have been the fragrance in the Caudalie cleanser, and I've had problems in the past because of this. But I also think the specific betaine allergen might have been the cause, or added to the flushing, and I checked on the link - it sounds very nasty!
I broke my rule of always patch-testing new products, and I won't be doing that again!
Yesterday I chucked out the Caudalie and bought another Laura Mercier foaming cleanser - success!
No flushing today, so I will stick with what works.

A quick update - my skin has now stopped flushing, about three weeks after I had to start my antibiotics again. The doctor has given me doxycillin for my rosacea, with instructions to take a course of two or three weeks, or even longer if necessary, whnever flushing becomes troublesome. I would much rather not take antibiotics in the summer as they increase sun sensitivity, but as I stay out of the sun generally it wasn't a problem.
But - what a hassle!
In future I'll stick with what I know works. Experimenting has proved far more expensive than my Laura Mercier!

A quick update - my skin has now stopped flushing, about three weeks after I had to start my antibiotics again. The doctor has given me doxycillin for my rosacea, with instructions to take a course of two or three weeks, or even longer if necessary, whenever flushing becomes troublesome. I would much rather not take antibiotics in the summer as they increase sun sensitivity, but as I stay out of the sun generally it wasn't a problem.
But - what a hassle!
In future I'll stick with what I know works. Experimenting has proved far more expensive than my Laura Mercier!