Black is probably the common, red is probably the run winding and blue probably the start winding but those are just guesses. Look for markings on the side of the compressor body like R,S,C to verify. The first thing to do is disconnect those wires preferably not at the compressor end and noting which wire was connected where so you can put it back and measure between them with an ohm meter. Between C and R you should measure a few ohms. Between C and S you should measure a few more. If either of these measurements gives you 0 or open then the compressor is shot and needs to be replaced. The next thing to do is to figure out how the compressor is started. There will be a capacitor or capacitors involved which in the pictures you have posted appear to be integral to the compressor itself i.e. inside the hermetic can. That, I would think, would not be something that the manufacturer would do as capacitor failure is pretty common (just happened on my heat pump - as soon as the weather turned cool and we turned it on) and is an easy fix - if you can access the caps. But perhaps on these $130 units it is cheaper to throw them away and buy a new one as I can't imagine getting a repairman to your house and having him fix something for less than $130. If you can access the capacitors use your multimeter to check that their capacitances are approximately as labeled. If the cap(s) check out then turn your attention to the control part of the circuit. There are several ways to start a compressor involving various types of relays (or not) so I can't comment on what you may actually have. In small, cheap units it's common to just have a run capacitor connected all the time between the R and S windings. In more sophisticated designs there may be a separate relay, a "start assist kit" may be installed or there may be a controller circuit board which manages the S winding connection. You will have to trace the wiring through from the black (hot wire) in the line cord through to the compressor. In a normal start cycle voltages, but not necessarily the same voltage, will appear immediately on both the start and run windings. After a second or two the voltage on the start winding may change but the run winding voltage should stay at close to line voltage (120). If you are lucky a wiring diagram will be attached to the inside of the cabinet somewhere. If there is one take a pic of it. This will make it much easier to give you better directed suggestions.
Don't know what a BCS is so can't comment on that.
One more thing: check for a thermal overload switch on the compressor body and make sure it is, if tripped, reset.