I think panniers are better than those lollipops, which didn't really stick out anyway. I use Bikebins and have horizontal reflective strips on the back of them. I think that motorists look at your back wheel when they should be looking at your main body. The panniers are about 8cm wider than I am, so 4cm or two inches either side but I think they buy me an extra few feet of clearance. I think that instead of fixing on my narrow back wheel, they fix on my panniers instead.
Anyway, the OP sounds like just another bullshit excuse for bad driving and impatience. It's never their fault.