How to remove magnet stuck inside driver head

Yes, but it is not without some risk.

1. You can try blowing high pressure compressed air across the opening of the driver. This likely won't work and if it does, the magnet may be blown to where you couldn't find it. You could aim it toward a draped sheet and hope to catch it.
2. You can do the following (more risky)
   1. Remove the endcap of the screwdriver. It is only a friction fit to an o ring in something very like a gas orifice.
   2. Hold the driver in a vise with padded jaws (preferably cork, but rubber should do)
   3. Remove the brass piece by using a straight slot driver in the slot in the top of the brass piece. Tap the piece smartly a couple of times to help it break free or you will chew it up. Get a good fitting driver to fit that slot.
   4. Using a pin punch (shown in photos) tap the magnet free from the driver toward the mouth of the driver
   5. The magnets should fall out along with some magnetic debris.
   6. Separate the magnets. Put the original driver magnet on a table with the shiny side down.
   7. Use a long steel rod, like a piece of a coathanger with a flush cut end to go down the barrel of the driver until the magnet grabs it. You can also just carefully turn the driver over while you hold the magnet in place.
   8. Use a bit or the punch (better) to return the magnet to its seat, tap it a couple of firm taps with a small hammer. it seems to stay in place.
   9. For more security, you might try dropping a small amount of iron filings down the tube from the other end along with a drop or two of cyanoacrylate glue This will likely bond sufficiently to hold the magnet more firmly.
   10. Screw in the brass piece (maybe with a bit of light duty threadlocker)
   11. Replace the endcap
3. Pics In rough operation order. The second pic is the screwdriver (you might need a wrench to turn it) the third pic shows the brass insert out. The 4th pic shows the punch in use, while the pics after that show you the punch , how it is to be inserted, and and how you tap with a hammer. the last pic shows the magnet pulled out.