If I try to add up neutrino masses (let's assume 1 eV rest mass equivalent each) to count as DM, do I use the rest mass or relativistic mass?
Use the total energy density since that is what appears in the Friedmann equation. Note that since the neutrinos are highly relativistic they behave more like "radiation" than "matter" in the Friedmann equations. You can call the total energy the relativistic mass if you like, but it is an outdated term. Nowadays people just use "mass" to refer to the rest mass - less opportunity for confusion.
It is known that the standard model neutrinos cannot account for the majority of dark matter. If you want to use neutrinos for DM they have to be heavy sterile neutrinos. (Sorry I don't have a reference for this off hand.)