  Electron beam fixed target experiments such as NA64 and LDMX use missing
energy-momentum to detect the production of dark matter and other long-lived
states. The most studied production mechanism is dark Bremsstrahlung through a
vector mediator. In this work, we explore a complementary source of missing
energy-momentum signals: Bremsstrahlung photons can convert to hard vector
mesons in exclusive photoproduction processes, which then decay to dark matter
or other invisible particles, such as neutrinos. We find that existing NA64
data can improve the leading constraints on invisible light vector meson
decays, while a future run of LDMX could improve them by up to $5$ orders of
magnitude. For the examples of a dark photon and a $U(1)_B$ gauge boson
mediator, accounting for meson decays substantially enhances these experiments'
sensitivity, especially to thermal relic dark matter of mass $m_\chi \gtrsim
0.1\ \mathrm{GeV}$.
