If you choose the timer route and want to tinker a bit, consider a reduced solenoid "holding" current. I don't know about the 252 solenoid in particular, but many solenoid accessories with a spring-loaded return mechanism will stay activated with a reduced applied voltage/current.
Apply full 14V to lower the gate. Once down, apply less than 14V to keep it down. Lowering a solenoid driven gate takes less than a second. Then, let's say you only need 10V to keep it down. Since heating power is proportional to voltage-squared, going from 14V to 10V drops power in half. This may be enough of a drop in heat to allow continuous solenoid operation. Your mileage may vary. Again, this would take some tinkering/experimentation but maybe 50 cents in parts (in addition to the timer) using a resistor or some diodes to drop the 14V to a lower holding voltage.
