We report on a novel bichromatic fluorescent imaging scheme for background-free detection of single CaF molecules trapped in an optical tweezer array. By collecting fluorescence on one optical transition while using another for laser cooling, we achieve an imaging fidelity of 97.7(2)% and a nondestructive detection fidelity of 95.5(6)%. Notably, these fidelities are achieved with a modest photon budget, suggesting that the method could be extended to more complex laser-coolable molecules with less favorable optical cycling properties. We also report on a framework and new methods to characterize various loss mechanisms that occur generally during fluorescent detection of trapped molecules, including two-photon decay and admixtures of higher excited states that are induced by the trapping light. In particular, we develop a novel method to dispersively measure transition matrix elements between electronically excited states. The method could also be used to measure arbitrarily small Franck-Condon factors between electronically excited states, which could significantly aid in ongoing efforts to laser cool complex polyatomic molecules.