Circuit to control a magnetic field, using a laminated core
I put 4 amps through the coil and measure 8 kA/m (10 mT) in the gap.
  This is the limit of the power supply I have, and the coil heated up
  quite a bit.


I'm assuming you are talking about DC current here. With 4 amps and 8 kA/m it sounds like you have several hundred turns and if those turns are producing too much heat your only feasible options are either to cool the winding or use thicker copper wire.


  Any advice would be greatly appreciated, even if it is an alternative
  set up


I might also add that the core gap under the coils does not make much sense.