There are volts, and there are amps. As a first approximation, you can think of volts as "pressure" and amps as "flow". The components represent the "plumbing" for the electricity. For your LED strips you need at least 5 and no more than 7 volts of "pressure" to make the required current flow through your system of electronic plumbing. 3 AA batteries provide only a bit over 4.5 volts when fresh, declining from there with use. So that is simply not enough pressure to make your plumbing work properly.
You certainly can get 200 mA, and a lot more, from 3 AA batteries, depending on the resistance of your "plumbing". In your case the resistance is high enough that 4.5 volts of "pressure" won't allow 200 mA of current to flow. But if you were simply to connect the two ends of the battery pack together with a wire (very low resistance) the battery's "pressure" would push several amps of current through that wire.
This is called the "hydraulic analogy" where the flow of electricity is compared to the flow of water in pipes. It is very basic and doesn't describe accurately all the things you need to know about volts, amps, watts, ohms, but it may make a useful picture in your mind so that you can understand what's happening in your system.
A battery rated at 3.7 volt, 800 mA-H, like your LiPo, is supposed to be able to provide 3.7 volts and 800 mA of current for one hour, or 3.7 volts and 1.6 A current for half an hour.... or 3.2 A current for fifteen minutes.... etc. The current depends on the voltage and the resistance of the load: this is Ohm's Law V=IR. And the length of time the battery can produce the current depends on the energy capacity of the battery and the magnitude of the current: 3.7V and 800 mA-H.