Saturating magnetic field in an inductor without heating
Here's one approach, based on the old DC controlled variable reactance technique. The point of this technique is to let you use your DC bench supply to saturate the core, while only needing some low power AC method to measure the inductance/reactance.

Put two separate, non-overlapping windings (with the same number of turns) on your core. e.g each covering 1/4 or the circumference. 
Connect them anti-phase in series, to your DC variable power supply.
Because these two windings are anti-phase, AC will be cancelled out.
Because the two windings are physically separate parts of the core, the DC will saturate the core region under the winding (making a virtual airgap).


You only need enough turns, so that your DC current can saturate the core. If you use 5 turns on each winding, and 10A saturates the core, then a 280 turn coil will saturate at 5*10/280Amps

Now you can put another winding/s on that you use to measure the AC inductance/reactance/transformer coupling with. 
You can measure the inductance of this winding with any other approach e.g inductance meter.
This AC winding doesn't need any particular number of turns, it just needs to be measurable for you. 

You will increase the DC current from your power supply until the inductance drops by what you define saturation as (eg to 1/2).
You are of course measuring DC saturating current. 

If it was a pure AC current it will be peaking at 1.4x the RMS value. So you might say that the AC.sat.RMS = DC.sat/sqrt(2). i.e if DCsat=10A, AC=7.1Arms. However saturation is not an absolute hard edge: it remains up to you to define how much saturation is the limit. When you are using AC most of the waveform is below saturation, so you might say that 9Arms is OK for you.