These coils will actually be sustainer coils. The intention being to keep the guitar strings sustaining as long as the player wants them to. It does this by tapping off the guitar's standard pickup output signal, amplifying it & feeding the signal into these coils...this creates a varying magnetic field...& the guitar strings are ferrous so react to the mahgnetic field.
Do you even need a magnet?

A magnet is required in the pick-up (the pick-up is actually a reluctance sensor) since the strings are not magnetic (i.e. they don't produce a flux). However, to vibrate the string surely only requires a varying magnetic field which could be produced with just a coil and a suitable pole piece. The effect of your magnet is to produce a strongly biased field - I can't see how that helps to produce the drive you require.