If this SJ cable is what looks like a heavy duty vacuum cleaner cord, then this is cord, not cable. Generally, cord can not be used for permanent premises wiring, and that is why you can't find staples for cord. If you were jumping between two pieces of machine, and that is what the instructions said to use, you may be able to get away with it (especially if there was a plug on the end or the distance was real short). But when you run it in or on the wall, it becomes premises wiring and must be a listed type of wire or cable for that application.
Is there any reason why you just can't run a 1/2" flex conduit the whole way and pull individual #14 wires through that conduit (typically the minimum size wire you can use)? Are there installation directions for this unit that are intended to apply to the US? Did they say to use a cord type conductor?
This16ga SJ cord isn't correct from the disconnect to the compressor. This must be #14 wire minimum and a wet area cable (SE, UF) or individual conductors with a W in them (THWN). The remaining wiring is kind of gray because you're going between pieces of equipment, but you still need to use an NEC legal wiring method if running this wiring in or on the building.