So far, ok.
Can you measure the current out of the remote starter input pin to GND? You probably can do this by putting a DMM in current mode, and letting it be the low-resistance path to GND. You need to know the peak current the transistor is going to handle before you can select the transistor.
I would increase R2 to keep the peak current into the base below 15 mA, and preferably at 5 mA.
What is the minimum low-input time the remote starter needs to operate correctly?
If R3 is connected directly to GND, it will not affect the C1 charging time calculations. With this change, consider R1 = R3 = 10K; R2 = 1K.
ak