You are describing a resonance in the scattering of protons on antiprotons. This is the measurement of the total crossection as found in the [particle data group][1] 

[![protonantiproton][2]][2]

No resonances are seen, so the answer is that no, there is no such resonance in the region of energy explored up to now, to go into the probability of its having an $e^+ e^-$ decay.

As  an example let us take the Z boson which [has a decay][3] into  three pairs of quarks, in principle, such pairs could form in the complicated phase space of the [proton][4] and antiproton, but they would have to be within the  mass of the Z , in order to have a probability to decay only  into an  $e^+ e^-$ pair. The measured cross section  reflects how improbable this is.

In contrast  [electron positron][5] scattering  has a large number of resonances.

[![elpos][6]][6]

which will have a decay channel into electron positron pairs.


This difference is not surprising, because protons and antiprotons are composite particles and their scattering products have to follow conservation of quantum numbers of their components, which happens by pairing quarks with antiquarks, at least three pairs. One pair could be a resonance from the $e^+e^-$ plot , decaying to electron positron, but there will be other products of the scattering.


  [1]: https://pdg.lbl.gov/2015/hadronic-xsections/rpp2014-pp_pbarp_plots.pdf
  [2]: https://i.sstatic.net/Mksty.png
  [3]: https://pdg.lbl.gov/2018/listings/rpp2018-list-z-boson.pdf
  [4]: https://profmattstrassler.com/articles-and-posts/largehadroncolliderfaq/whats-a-proton-anyway/
  [5]: https://pdg.lbl.gov/2019/reviews/rpp2019-rev-cross-section-plots.pdf
  [6]: https://i.sstatic.net/ZCxkz.png