Klugesmith wrote ...

This is a good opportunity to discuss wire size rating for short pulses of current.
The traditional approach is based on acceptable temperature rise from adiabatic heating of the conductor.

The important parameter of the current pulse is its "action integral", integral of (i^2 dt), in ampere^2 -seconds.
The A^2-s rating of a wire is proportional to the square of its cross-sectional area.
Some practical numbers and underlying formulas can be found in:
From their tables, an AWG14 copper wire could withstand 47,000 A2s without damage to thermoplastic insulation.
That's 217 A for one second, or 2170 A for 10 ms, or 21.7 kA for 100 us.

If your pulse were at peak amplitude of 200 A for the whole 10 ms, its I2t would be 400 A2s.
But if it's a half-cycle of sinewave, I2t would be only half as great: 200 A2s. Could be handled by an AWG26 wire or equivalent area of PCB track, if allowed to cool between shots.

For lightly damped oscillations, the action integral builds up with each cycle.
It's simpler to just figure the temperature rise, assuming that the entire energy from capacitor goes into heating the coil. You can look up the specific heat and density of Cu. As an unreviewed example, consider a single-sided PCB 4 inches in diameter, with 1-ounce copper foil, and 5% of the area etched away to leave a spiral coil. It would have about 2.35 grams of Cu. A 100 Joule discharge would raise the temperature by a tolerable 110 degrees C, dropping rapidly as heat is conducted into the laminate.

Thickening your tracks with solder would help, mainly because of the extra thermal mass, which is all there on 10 ms timescale.