Based on your comments I'd say you're on the right path on your own.
You already mentioned using the (ideal) gas laws. Beyond that there isn't a whole lot to consider in this problem. (assuming you want to do a cursory analysis of this situation, and not find exact pressure values or anything like that)
It's worth noting that water is approximately incompressible, so it doesn't build pressure quite like the gas does, but as long as the gas is pressurized the water will be at that pressure (and a bit higher the deeper you go, see hydrostatic pressure).
So the basic concept is apply a weight > deform the container > reduced air volume means increased pressure in the bottle due to ideal gas law.
Your rewording of the question is a bit more complicated. The effectiveness of transferring pressure between outside and inside depends quite a bit on the material of the bottle. Depending on the material, it will deform more or less when you apply a force to it.
A weak material, like a balloon very easily deforms, so it's very easy to apply a force to the outside and increase the pressure inside. Something like a steel container doesn't work as well. When you apply a force it does deform a bit, which would increase the pressure; but it barely deforms, so the volume barely changes and pressure barely increases.
Tanker cars that carry fluids at normal temperatures and pressures are generally designed to withstand the outward pressure that the fluid exerts on the walls. The construction is usually cylindrical or ellipsoidal in shape. These shape help to distribute the hydrostatic forces evenly and eliminating sharp corners reduces the likelihood that pressure stresses will concentrate over small areas.
But if you empty a tank of the same construction, seal it and provide any means to evacuate or reduce the pressure within, there are two factors that can lead to sudden collapse. (1) the vacuum pressure force is in the opposite direction that the normal hydrostatic forces impose on the tank and (2) the vacuum forces could conceivably reach greater pressure than what the hydrostatic forces reach (up to 14.7 psig). If the tank uses external gussets that reinforce the strength of the tank, the gussets no longer assist in supporting the tank wall under vacuum since they act to support in the opposite direction.
But to answer your primary question, all it takes is one minor deflection, fold in the tank wall, the even distribution of pressure is lost and all of the sudden stress is concentrated at that one point. This leads to rapid escalation of the collapse.
Best Answer
Since you know the force on the bottle (roughly 200N), you would have to get an approximate area over which this force is distributed. You could try by spreading some ink on either the bottle or the weight to estimate the contact area. While this is not completely correct (the wall of the bottle does redistribute the pressure on the outside to a larger area on the inside), it should give a reasonable estimate to maybe a factor of two or so (but that's a guess of mine, of course). If you want to do it right, connect a long plastic pipe to the bottle cap and measure the height of the water column that gets squeezed out. That's a direct and fairly reliable pressure measurement. Lacking such a measurement, you could estimate from the squirt height. I am sue there are better ideas, still, but these are the easy ones that I can think of of the top of my head.