Alan, I prefer the term invariant mass over rest mass and your question provides an example motivating that preference.
The terms in a relativistic Lagrangian for some QFT must be Lorentz scalars; they must be invariant under Lorentz transformations, i.e., they must have the same value in all reference frames.
When the Higgs field has a vacuum expectation value, the Standard Model Lagrangian can be written such that the matter and gauge fields have associated "mass" terms. These terms are, of course, Lorentz invariant.
So, in this way, it is clear that the interaction results in an invariant mass, one that is the same in all reference frames.