I had the chance to fire up my ANSYS simulation and thought you would enjoy some examples of what it can do. Attached is an animation and plot of a nominal induction launcher design with the following parameters. Note that this is a time harmonic solution (not transient) with 10 identical coils and an armature that is longer than the "barrel". A case which would be great for efficiency, since all of the coils are running at the same frequency and all doing work on the armature. So a bit of a simplified launcher.

Armature diameter: 25mm
Armature wall thickness: 3mm
Armature material: Aluminum
Armature weight: 200g

Coil number: 10
Coil cross section: 20mm x 15mm
Coil wire gauge: 10 AWG
Coil turns: 45
Coil drive current: 1kA

With this model I can sweep all sorts of variables like armature-coil gap spacing (one I knew you were keen on), drive phase number, drive frequency, etc. etc.

The animation shows the product of the azimuthal current density J and the radial magnetic field density B which has units N/m^3 (if you do a volume integral you get total force on the armature). The plot shows the total armature force vs drive frequency for gap spacing of 1, 2, and 3mm. As you can see, a smaller gap is better, but the gains aren't astronomical, more on the order of 10 or 15%, at least for this design.

I would be happy to plug in your numbers and run a similar simulation sweeping whatever variables you are interested in. Just let me know!

That looks like within about 10% of prediction which is pretty good. Certainly close enough for initial tuning. Did you account for the tolerance of your capacitors?

Have you seen this video, about what's inside a similar AC sync indicator?
It includes current measurement, and a demonstration of fixed 12:00 and 6:00 as I'd suggested.
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How do these gate waveforms look? Is the rise time too slow?

I have no idea what rise time is because scope time/division is not listed.  Oh, after noticing title saying 100kHz, I can then deduce scope is at 2us/div for first capture.  Listing both volts/div and time/div for all traces would be helpful.  Or amps/div for current.
Rise time is probably fine.  Some rise time is needed to provide dead-time between turn-off and turn-on.

Also, do you know why I sometimes get the noise on the primary current waveform at the switching transitions but sometimes not? In this case it might be relevant to fix the noise because it lines up with some ringing/noise on gate turn off.

Presuming UD2.7 or similar driver, phase lead does not function well at low current.  Once current builds, phase lead works better.  Phase lead makes switching at proper time slightly before zero current which reduces spikes.

I'm building a small half bridge coil. How do these gate waveforms look? Is the rise time too slow? and what about the ringing on turn off?
I've got 5.1 ohms gate resistors and 0.82 ohm turn off gate resistors. these values worked well for lower frequency coils but I might need to tweak them.

I'm scoping the low side gate because if I scope the high side, I get the primary current distorting the gate waveform somewhat.

I've also included shots of the bridge output with phase lead tuned as well as possible at 200A primary current. I'm planning to run it up to at least 600A But that was as high as I could get it with my isolation step down transformer at 40V AC in.

Also, do you know why I sometimes get the noise on the primary current waveform at the switching transitions but sometimes not? In this case it might be relevant to fix the noise because it lines up with some ringing/noise on gate turn off.

Thanks, Benjamin