Is there any efficient chemical method of eradicating just those Berberis shrubs?
I haven't tried it on barberry but round-up is typically a good candidate for targeting a specific individual. Apply it to the leaves, it kills the roots but breaks down quickly in the soil. So if you keep it off of neighboring plants as you apply, it won't harm them. Depending on the size of the plant and the density of the neighbors I will either spray or carefully brush it on the leaves. This works well for European buckthorn.
I have removed quite large berberis with just a shovel. They do not have a thick tap root system like junipers. If you trench around the plant and dig down you should be able to pop it out. A pair of secateurs or bypass pruner will help you cut any thick roots from the cedar.
Backfill the hole with compost or topsoil to finish the job and water the area well for a week to make up for broken roots.