My plan was to use an off-the-shelf audio amplifier chip and feed it a sine or square wave from a function generator, the frequency matching the string's tuned frequency.
If you force a string to vibrate with an external sine wave excitation, the string will vibrate at the frequency of the function generator; it will not vibrate at its own natural frequency. And if the function generator frequency is poorly matched to the natural frequency, the string won't vibrate much. The response away from resonance depends on the Q of the string.
If you are building an electronically-played guitar, then you need to pluck the string with a current impulse, like a musician does with a pick. Then the string will vibrate at its natural frequency.
If you want to build a sustainer, then I would just amplify the pickup signal and send it to the electromagnet. Essentially, this would be some controlled amount of positive feedback.
If you wind a solenoid (bunch of turns around a straight core) that s a pretty long magnetic path. You will have field over a fairly large area. That may or may not matter in your application.
Your coil has both resistance and inductance. You are running at audio frequencies so the inductance is not a massive issue. I would aim for a resistance on the coil *below* 8 ohms to allow for some inductance. You will have eddy current losses in a simple core, so the net result is going to be a bit exciting to try to calculate exactly.
As you sit there counting an enormous number of turns. Consider that 4 ohm output amps would require half the number of turns ....
I'm thinking gapped toroid with the string in the gap. This may also increase isolation between the strings. I wonder if you could get some DC current through the strings if that would help the sensitivity.
As for the impedance/number of turns issue, a transformer may help things in that department.