Welcome to the forum.
Unfortunately it is not that simple. A capacitor big enough to support a 900mA current draw for 15 seconds without significant voltage droop is going to be too high a value to consider. There is also the issue of how the 5V power supply would react to having to charge a capacitor that big at turn-on (when it would look like an effective short circuit).
You don't say what is generating your 5V supply but it sounds like it is unstable when faced by the hostile environment of a car electrical system, which is full of noise, spikes and voltage variations. It isn't too surprising that the screen drops when the starter is operated because that draws several hundred amps and can cause a significant voltage drop depending on the condition of the battery. Windscreen wipers indicate that the power supply is susceptible to electrical noise rather than voltage drop - unless there is a grounding problem somewhere.
This is a case of finding the cause rather than trying to apply a, very large, sticking plaster.
Edit: Sorry, I misread your post slightly. You say that you don't want the screen to run for longer than 15 seconds on the capacitor, but that implies that you do want it to run for something in the same order of magnitude, so the above comments still apply.