I think that there is no such thing as a bream proof rig, but how you feed can be the trigger.
Small silver fish and bream are attracted to heavy baiting, especially with particles, pellets and groundbait. The more you bait the more interested they are.
Bream tend to be vacuum feeders, they get onto food and they clear it up, and that includes the hookbait, no matter how it's fished, bottom baits and low lying pop-ups are especially vulnerable.
To avoid them feed less, or try boilies only.
Don't put any free bait in, and fish just PVA bags or stringers, or fish boilies where although you may put 100 (example) in, they are not so close to invite or become a feeding trigger.
The other thing is bream tend to avoid the margins, preferring to feed out in the main lake. They can have particular patrol routes!
Now saying that, I used to fish Ardleigh reservoir and you could catch bream and then the carp would come before the bream pushed them out again. I caught both bream and carp on baited spots, yet on Alton, I never caught a bream despite using loads of particles.
My rigs on Ardleigh and Alton were simply line aligned knotless knotted snowman baits on braid rigs, nothing special.