How to power small solenoid with a 3.7V lithium battery pack
Maybe consider using a buck regulator to convert the 3 volts to 1.5 volts. Given that they convert voltages with usually a power efficiency about 95%, the 1.333 amps required at 1.5 volts (2 watts) translates to just over 701mA on the 3V side.

However, given that you appear to say that the battery pack can supply only 667 mA, you might be screwed unless there is a guaranteed lower operating limit on the solenoid that is smaller than the numbers provided in the question.

If not either add a battery just for the solenoid circuit or make the basic battery bigger. I only have your data to go on.