Electronic – Calculating current through multiple capacitors in parallel

The part's datasheet says "Unconditionally Stable with 2.2μF Ceramic", so I think you're good.

(Ceramics have lower ESR than tantalum or aluminum electrolytics.)

Notice the note about increasing the capacitance if you decouple the BYP pin.

An efficient way to charge high capacitance capacitors with a small current is to use an inductor. i.e., you see the technique all the time in voltage converters. i.e., charge an inductor field by placing a voltage across it, thus building up the current in the inductor. Then, place the inductor in series with the capacitor that you want to charge. I.e., high speed switching. The inductance in the inductor will limit the current, while the energy stored in the inductor field will cause the voltage of the inductor to change to meet the voltage in the capacitor that you want to charge. i.e., at the point where you stop charging the inductor, and place it in series with the capacitor, the inductor becomes a current source. You probably remember hearing in your first AC electronics course that capacitors resist a change in voltage, and inductors resist a change in current. These characteristics make them a perfect combination for solving your problem...