  We systematically study a set of strongly polar heteronuclear diatomic
molecules composed of laser-coolable atoms for their suitability as sensitive
probes of new charge-parity violation in the hadron sector of matter. Using
relativistic general-excitation-rank configuration interaction theory we single
out the molecule francium-silver (FrAg) as the most promising system in this
set and calculate its nuclear Schiff-moment interaction constant to
$W^\mathrm{FrAg}_{SM}(\mathrm{Fr}) = 30168 \pm 2504\mathrm{a.u.}$ for the
target nucleus Fr. Our work includes the development of system-tailored atomic
Gaussian basis sets for the target atom in each respective molecule.
