Keeping a consistent supply voltage while driving a solenoid from battery
First part of your question can be answered experimentally. Charge those capacitors to 15V, disconnect them from the 15V source, and apply them to the solenoid. If the solenoid operates acceptably then the capacitors have enough capacitance and you're good to go from that perspective. Otherwise you'll need more capacitance (or more voltage!). 

Secondly, you may need some way to isolate the drain of the solenoid from the boost regulator output to prevent it from dragging down the input voltage. The easiest way to do that is to use a resistor that is high enough value to prevent overloading the source and low enough to allow the capacitors to charge in a reasonable length of time. Suppose you allow a 25mA charge current, which might be 75mA at the batteries. That would be a 600 ohm resistor, so let's try 680 as a standard value. Power dissipation should not be much but if you use a 1/2W resistor it can't get too hot even if you leave it on. 

The time constant of 680\$\Omega\$ and 4400\$\mu\$F is about 3 seconds, so after about 10 seconds or so the capacitors will be fully charged.