The prospects are presented for precise measurements of the branching ratios of the purely leptonic Bc+ → τ+ντ and B+ → τ+ντ decays at the Future Circular Collider (FCC). This work is focused on the hadronic τ(+) → π(+)π(+)π(-) nu(tau) decay in both Bc+ → τ+ντ and B+ → τ+ντ processes. Events are selected with two Boosted Decision Tree algorithms to optimise the separation between the two signal processes as well as the generic hadronic Z decay backgrounds. The range of the expected precision for both signals are evaluated in different scenarios of non-ideal background modelling. This paper demonstrates, for the first time, that the B+ → τ+ντ decay can be well separated from both Bc+ → τ+ντ and generic Z → bb(-) processes in the FCC-ee collision environment and proposes the corresponding branching ratio measurement as a novel way to determine the CKM matrix element |V(ub)|. The theoretical impacts of both B+ → τ+ντ and Bc+ → τ+ντ measurements on New Physics cases are discussed for interpretations in the generic Two-Higgs-doublet model and leptoquark models.