It is good that as a high school ( from your profile) student you have an interest in particle physics, and I hope this continues to university years. I am treating your question as a thought experiment, and will comment accordingly:
focusing on J/psi pair production through proton-antiproton collisions at a range of momenta.
The J/psi meson has a mass of 3.1 GeV ( in a system where c=1).
By j/psi pair production you must mean : by looking for e+e- pairs produced in the antiproton proton scattering.
The decay modes of the J/psi are mostly to hadrons, but there is a substantial fraction that goes to e+e- and mu+mu- ( ~12%).
to collide a beam of antiprotons with protons in a bound state
There does exist a bound state of proton antiproton, called protonium, but when colliding a proton and an antiproton at high energies they do not bind because their respective energies are too high. Protonium may exist as a resonance in the decay products, but this is not what you are describing. You want a proton antiproton collision to give, among the products of the collision, lepton antilepton pairs whose invariant mass may show the J/Psi resonance.
The linear accelerator I have access to has a range of momenta from 0.5 GeV/c to 10 GeV/c,
To find the energy available for particle generation after the collision, one has to find the input energy. The input energy of each, proton and antiproton is given by adding their four vectors , an example can be found here .
The incoming energy must be higher than the mass of the J/psi as energy, so there is a threshold energy incoming for seeing J/psi production. An experiment needs high energies to have a probability of generating such a heavy resonance as the J/psi. My impression is that only at the tevatron energies this experiment could be realized.