There are some questions. If this is a stand alone bench with stand alone welding equipment there is no problem having 120, its done all the time. You are looking to avoid an alternate pathway for welding currents is correct but any metal object exposed to power needs to be grounded. Only exception a guy could make is to feed it from a gfci circuit. If its not grounded,, and this does NOT mean a ground rod it has the potential to become energized, say from a pinched wire from a tool cord, standing 120V, no fault return.
Put the work lead from the welder on the bench, not a problem. In my own case have a steel building, benched welded to the floor and bonded to building with a rod at which point have common work grounding for multiple welders. I faced thidilemmama at one point, equipment all over I finally made some changes but supply the bench with a wiree3 circuit preceded gfci, the bench IS grounded just not with a conductor from the electric circuit, 2 benches in this scheme but ieliminateded the possibility of ground wire from being used as conductor for welding current.
Main thing is not to get confused, ground rods are nevesubstitutetute for an equipment ground. Ramble, probably clear as mud, jump in without hurting my feelings, ha