Let me first briefly explain how OpenSprinkler DC works: it uses an internal boost converter to jump the input voltage up to about 22VDC, and that voltage is charged to a capacitor, which is then directly to the valve to energize the valve (i.e. provides impulse current). The controller then lowers the voltage to input (about 7.5V) to provide holding current. This imitates how the valve behaves under AC current, hence OpenSprinkler DC is able to operate AC valves using DC only voltage.
In theory, whether this works for pump start relay or not depends on the electric spec of pump start relay. But we’ve tested a number of pump start relays on the market and they all seem to work pretty well with the same mechanism. We haven’t tested your particular one and that’s why we need to get one to test in house and see what may be happening. The resistance seems all right so I don’t see any obvious reason why it wouldn’t work. You said QP2 and QB2 are blown? I don’t think this has to do with RP2 — RP2 is a 100K resistor and you would have to apply a very high voltage to actually short that resistor (also resistors generally fail open and not fail short). You can send a support ticket if you suspect some components on the circuit board are blown.