Here's the dark side of replacing caps. Switching power supplies often work by stuffing pulses of current thru inductors into caps from a higher voltage. Primary failure mode of caps is increased series resistance, ESR.
When that resistance increases, there's a step in voltage across the cap when the current is being stuffed in. The regulator is looking at average voltage, so it reduces the output voltage slightly to make the average voltage, including that step, to be the desired output.
Over time, that step gets bigger and bigger. You don't see any symptoms until the step gets big enough to disrupt the circuit.
By the time the system shuts down, damage may have been done.
Had one monitor where the system processor had been fried by the peak voltage. Had 19V spikes on a 5V supply.
In another, there were totem-pole fets driving the transformer for the backlight. One of them had been damaged and had high leakage. The backlight came up and ran for a few seconds until the other fet overheated and blew the fuse.
That's why it's often a good idea to change all the caps at once. If you're lucky, and most are, you caught it before permanent damage was done.
So, back to your problem... Most any fuse of the correct current rating should work for more than three seconds. Yes, the right fuse is best, but fix the circuit first.