From the previous mines thread IIRC shields are not active against mines.
As for boarding parties...... now there's a good idea
As or sweepers treating large mines as 3 targets....nah. It would be better if there were large warheads to justify larger mines, say something that could seriously dent an organic kamikaze minesweeper. 400 damage and 15 - 20kt space requirement, expensive to build of course.
Allegiance subverters may work, it could trigger a combat though. So every ship that gets subverted then has to fight the rest in the minefield (allowing them to kill the mines with point defense and projectile (?) weaponry.