In my opinion, it is an unannounced failure of the ILS system. If there is sufficient time while on a vector or established in a procedure turn for the approach to be re-briefed, then by all means brief the changes and continue, however once established inbound of the FAF I would execute the missed approach. I personally think that an examiner may take issue with significant changes to the instrument approach profile while inside the FAF. My concern is that, as a single pilot in the IFR environment, descending towards the ground/airport inside the FAF, your attention is taken away from flying the aircraft to re-brief certain items. If the glideslope has already guided you below the MDA for the G/S Out Minimums, then instead of immediately going missed approach you looking down at the approach chart during a critical phase of flight.

I think it really all comes down to what you brief, but on a checkride I would be particularly hesitant on rushing to make changes like these. Timely, is again not the same as rushing, bottom line once inbound from the FAF any Lateral/Vertical Guidance Failures should immediately dictate a missed approach.

Also, as a side note advise ATC as soon as practical of the G/S Failure.