Interesting question - My comment is just an "opinion".  Going beyond the design specifications is a "no-no".  However, upset conditions happen and sometimes the vessel is subject to above-specs conditions.  That is why the Code has built-in safety factors.  One approach might be to find the weakest component and go to 90% of yield for maximum stress (or similar such limit).  Concerning the time line, even one second of overpressure may damage the vessel.  How about the cycling/fatigue issues?
If I were the owner of the equipment I would say 'don't do it'.
Good luck.

In my opinion the answer to your question is in a small word that appears in AD-121.3: 'The operating pressure is the pressure at the top of the vessel at which it normally operates'.
This means that abnormal conditions of operation need not be included in the design pressure, but of course provided some conditions are met:
- the abnormal condition must not occur under predictable or repetitive conditions (otherwise it becomes normal): so only an accidental condition may be defined as abnormal
- as always with accidents, an analysis of the accident issues should be conducted before the plant/vessel is again operated: this analysis might lead to the conclusion that no permanent damage was incurred by the vessel (the procedure suggested by GatorOne is a starting point), or that repair is required (and of course also that some operational precautions should be adopted to avoid repetitions).

prex

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There is a clue to this question in ASME Sct III for nuclear components. The design philosophy for Class 1 vessels in NB-3000 is very similar to Sct VIII-2.
For Service Levels that have a low probability of occurrence Sct III allows higher stresses. E.g for Service Level B (which used to be called "Upset condition")the primary stress intensity (and hence the pressure) can be 10% higher than for the design condition.
The details of this become pretty hairy and I doubt that you can used it for your case.

I don't know how you could get there.  It is a requirement (except when using Code Case 2211) that the ASME Section VIII, Div 2 vessel have a pressure relief device with a set pressure not greater than the MAWP.  So your relief valve, rupture disk, etc would relieve if you had an overpressure.  Now if you use Code Case 2211, Pressure Vessels with Overpressure Protection by System Design, you are required to make sure, via hazop, etc that there are no circumstances in which pressure in the vessel can be reasonably expected to exceed the MAWP.  So I think the answer is no, at least from an ASME Code perspective.

I believe that the prevailing concept of Div 2 is that "Don't exceed MAWP". But in practial reason it doesn't seem to mean in absolute sense. When you do hydrostatic test, the pressure is normally higer than MAWP. And the pressure relieving device has a little tolerance. Also during pressure relieving due to over pressure, the pressure will be higher than MAWP. These are described in the Code.