Thanks for the help everyone, I ended up manually making the spreadsheet, with capacitors ranging from 7.5 to 82g and 16mm to 40mm diameter. Basically the higher voltage the better, within a capacitor family. Some capacitor families go up to 550V, but the most energy dense families only go up to 450V. The next largest factor is the connection style, in that snap-in capacitors are much lighter for a given capacity than either screw terminal or radial leaded capacitors. It also seems that smaller diameter capacitors had a higher energy density, but length had a fairly small effect. Manufacturer basically doesn't matter as their high end capacitors all have the same capacitance for a given case size and voltage.
Because of this, I ended up choosing the smallest diameter and longest snap-in capacitor available, the 22mm diameter, 50mm long, 400V 330uF United Chemi-Con ESMQ401VSN331MP50S, and am planning on putting 3 of them in parallel. They don't list the mass of their capacitors, but based on TDK's mass for the same size it is 2.66 J/g. The highest energy density I could find was 2.99 J/g, but that and several other capacitors at the highest energy density were not stocked anywhere.
In addition to their capacitance, all capacitors have additional 'stray' components that may have to be taken into account if the circuit is sensitive to them. These include leakage, equivalent series resistance ESR, series inductance, and dielectric relaxation, not to mention the ability to degrade with temperature and/or age.
Different technologies of capacitor will have different strengths and weaknesses. For ESR, alli electrolytic is highest, followed by tantalum, with plastic and ceramic the best. The inductance generally follows the size and lead length. For leakage, alli electrolytic is worst, though usually not a problem in decoupling applications, you just avoid using them for long time delays in a 555 circuit.
For decoupling your circuits, it is important to use appropriate sized capacitors in the right places.
Most op-amp circuits will be working with such low frequency components that almost any capacitor can be used for supply decoupling, as long as the leads are kept short, alli electrolytic will be fine.
Very fast op-amps, for example video capable, as well as digital circuits, will need small value, low inductance, short lead decoupling capacitors very close to the ICs. In addition, the board will also need bulk decoupling, an alli electrolytic is fine for this.
Best Answer
Each battery cell has an internal series resistance and when you stack up the batteries, these series resistances add up, too, limiting the current you can draw. You can load a capacitator with that limited current. The time it takes is not relevant for the function of your coil gun.
When you trigger the coil gun, the capacitator is discharged very quickly at a very high current. It can supply this current because its own series resistance is much lower than that of the battery stack. The height of the current peak is practically only limited by the inductance of the coil you use.