If we assume that the converter can work to zero voltage (which is actually far from the case), then 235 uF @ ≈300 V is just 0.07 C or A*s, or 21 W*s. For a typical PC load around 200 W it can provide only 0.1 seconds of working. Yes, there are more energy storages, as ThermallyFrigid mentioned above, but we are still talking about fractions of a second. So I completely support this opinion: if this is critical to you, you need a battery backup.
But if you are interested in exactly the question of why two identical configurations behave differently during mains power "fluctuactions" (provided that they operate under the same working load!), then I would recommend checking and comparing the output capacitors on the PC main power rails: +12, +5, and +3.3. But it may also be that the supervisor are simply more sensitive to undervoltage conditions, try to find its datasheet and check it.