The magnetic field depends directly on the amount of current through the coil. To overcome the reactance (due to inductance) of the coil, use a high voltage. Follow that with a current regulator set to the value that gives you the 2-5T field. The higher the voltage used, the faster the current will rise to the desired value.
Firstly, I do not think you can use a neodymium magnet as a projectile.
Coilgun
A coilgun (or Gauss gun, in reference to Carl Friedrich Gauss, who formulated mathematical descriptions of the magnetic effect used by magnetic accelerators) is a type of projectile accelerator consisting of one or more coils used as electromagnets in the configuration of a linear motor that accelerate a ferromagnetic or conducting projectile to high velocity.
And you should be more focused on the kinetic energy of the projectile as it leaves the tube. A 400 farad 2.7 v capacitor stores 1450 joules of energy, which given a efficiency of 1 percent translates to 14.5 joules of projectile energy. Plugging this in to the kinetic energy equation \$E = \frac{1}{2}mv^2\$, gives us a muzzle velocity of 170 m/s for a 1 gram projectile.
Now let's come to the question of force. The force depends on how much power you can pump through the coil. And this minimum force will be huge, because the projectile must reach muzzle velocity by the middle of the coil. (The coil must also be discharged by this time, or the projectile will slow down) For a one gram projectile, this will be 729 newtons or 73 kgs of force, assuming a coil length of 4 cm. (You can calculate acceleration from this equation - \$ a = \frac{v^2}{s}\$ where s is the barrel length and v is velocity)
And what does this mean? You need a strong projectile, definitely. And your capacitor must provide a vast amount of power. Taking the above parameters, the time before the projectile hits the middle of the barrel is 0.23 milliseconds, which you can calculate using the kinematic equation \$x = \frac{1}{2}at^2\$. Dividing the total energy by the time gives us a power requirement of 6.5 MW to be discharged. That's right. 6.5 MW.
With a 2.7 volt capacitor, the resistance must be below 1 micro-ohm. Definitely not possible. With a 400 volt capacitor, the minimum will be 24 milliohms. This is possible.
Now, your question specifies you are not interested in maximizing velocity. In that case, you can go through these calculations for your specific use case. The wire gauge depends on the current going through the wire and the voltage. Once you have that you can calculate the number of turns needed, and that gives you the resistance of the coil. You can add this to the resistance of the capacitor and the diode to give you the total resistance. This must be lower than the minimum resistance you calculated.
And of course, exercise caution. 1500 joules is a lot of energy.
Best Answer
The V700 should be V = 700*I and similar for the second source, but that source on the right should be polarized in the opposite direction. One other thing is that the air gap reluctance/resistance is much larger than the core and bar so you can ignore them without too much of an error. If you include them them for your circuit's resistances to be as they are then the coils would need to be at the edge of the horseshoe shaped core. If not then split up the core resistance into three parts with two of the resistors between the coils and the airgaps. Lastly, if any of the bars/cores are permanently magnetic then you must add a source in series to represent the magnetic polarization. For example a horseshoe magnet would need no coils due to its approximately permanent magnetization. Oh, reluctance is length/(permeability*Area) where permeability is the property of the material which gives your resistance values.