First of all, here considering the mass defect as the binding energy between an electron and the rest of the atom; i.e. taking the mass of nucleus as it is. That since nuclear mass defect is present even before an electron comes to the respective ion. And asking specifically on electron, since contrary to nucleons, it is not a composite particle.
When imagining mass defect of an atom naively, it is some negative mass that is either distributed among the positive masses of constituents of the considered atom (nucleus and electrons), or it is living separately in there.
When the mass of an electron is measured at such a system (i.e. residing in an atom), it is possible to imagine that the measured mass is affected by the mass defect; like being somewhat decreased due to the mass defect of the atom.
I expect that this is something that is routinely considered when measurements of the electron mass are done. And that either some corrections are applied for it, or it is supposed (and hopefully tested) that it has no effect on the measured electron mass.
UPDATE: Since the question is apparently unclear, notice please, that I know that the electron-mass measurements are based on spectroscopy.
The thing is that the spectroscopy measurements are based on atoms/ions that have some binding energies. And if a measured value is related to mass of electron, then it could be affected by the binding energy if the binding energy affects the effective electron mass within the measured system.
Stating that the formulas used for $m_e$ derivation do not know about binding energies, is alike stating that the binding energy does not affect the effective value of $m_e$, etc. within the bound systems. If it is that way, and it is checked by measurements, then OK.
Best Answer
It's correct that the mass of a hydrogen atom is different from the mass of a proton plus the mass of an electron, but this doesn't matter since this is not how the current value for the electron mass was measured.
It is determined from the hydrogen atom by measuring the Rydberg constant, $R$, and the fine structure constant, $\alpha$, then using the equation:
$$ m_e = \frac{2Rh}{c\alpha^2} $$