Numerical tolerances are hard, and the answer very much depends on the solver and methods you’re using. So, you might get a more informed answer from Gurobi support.
What you might try is solving the LP without the SOC, checking if the SOC is already satisfied (up to whatever tolerance is meaningful to you), and if not, add the SOC constraint and re-solve.
I do not want to obtain an exact solution or compute duals
Minor point, but all SOC solvers I’m aware of compute duals either explicitly or implicitly as part of solving the problem.