If you want to apply a force to the coil structure in the model, then you have to go with Transient Structural.
When you say the force input is available in the time domain, I assume the 4 kHz refers to the sampling frequency. A minimum of 10 samples per cycle are required for transient time history loads, so you should not see frequencies in the data above 400 Hz. An FFT on the signal will show you how much frequency content is above 400 Hz.
If the moving mass is easily calculated, then you could convert the force to an acceleration and apply an acceleration load to the mass and reap the benefits of using a linear analysis method called Random Vibration where you could take the PSD of the acceleration load and apply that to the model. I have some experience converting a time-history signal into a PSD so if you want to attach a zip file to your reply with a spreadsheet or text file of the force time history, I will take a look at it.
Though you say "I'm not looking for the stress at a specific frequency", once you have that linear system, it may be worth looking at a Harmonic Response, since it quick to do and you may learn something useful.