Does the hydrohelium dication exist?
I am not sure how to elaborate much beyond saying that yes, there are bound states of $\ce{HHe^{2+}}$. I cite three papers below which give numerical calculations of various states, ref. 1 gives calculations for the lowest 20 states of the system.

It appears to be that the 1s$\sigma$ is not a bound state in this system, but the 2p$\sigma$ state is bound. I would expect there are some other bound states as well.



References:

[1] Winter, T. G., Duncan, M. D., &amp; Lane, N. F. (1977). Exact eigenvalues, electronic wavefunctions and their derivatives with respect to the internuclear separation for the lowest 20 states of the HeH2+ molecule. Journal of Physics B: Atomic and Molecular Physics, 10(2), 285.

[2] Laaksonen, L., Pyykkö, P., &amp; Sundholm, D. (1983). Two‐dimensional fully numerical solutions of molecular Schrödinger equations. I. One‐electron molecules. International Journal of Quantum Chemistry, 23(1), 309-317.

[3] Buehler, E. J., Gooch, E. E., Dial, J. L., &amp; Knudson, S. K. (2000). Semiclassical energies of low-lying states of one-electron diatomics. Chemical Physics, 253(2-3), 219-230.