A few thoughts on the subject.
A utilization of 1.2 for a bending check is high but does not necessarily mean failure. It would however make me take a second look at the analysis.
As I understand it, the loading is know with confidence. That is a good start.
Second question is the geometry, span width, simply supported or more spans?
Third question, material im the beam. Since it is an existing beam, is the material known with certainty?
Then I would make an analysis using the loading with no load factors. Does the result keep me below yield stress? If it doesn't do that for simple bending with elastic section properties, this won't work.
The same analysis with plastic properties, what utilization does that give? My assumption is that both of these will be fine since you have 1.2 and that can be the result of a load factor.
You say that you have the exact weight of the equipment. How is the equipment installed? Could there be any dynamic amplification in this?
The factior 1.2 can indicate different problems depending of the exact context. What exact load factors have you used? What is your safety level?
As I understand the question, you don't meet the code, you wonder if it is dangerous? Probably not, depending on the exact circumstanses. But even if it isn't dangerous (it will probably not fail), why should you take the risk/responsibility?
Speaking in general terms:
I have done this type of analysis using non-linear FEM-analysis. I would model the critical beam with shell elements to get as accurate stress distribution as possible within reason. Then I would start with nominal load and increase the loading to failure.
When I know how it fails, when it fails and how sudden the failure is, then I make my conclusion/recommendation. But I would also explain to the the client that the structure does not meet the requirements in the code. The latter part can be a pedagogic issue if you don't have a professional client.
Thomas