Unfortunately, a current transformer is built as a transformer, which means it has very high inductance, high permeability, so stores very little magnetic energy for its weight and cost. In fact, a transformer specifically designed as a 'current transformer' is going to be even higher permeability and lower energy storage than your average power supply transformer.
A flyback transformer, although it has 'transformer' in the name, is built on a low permeability core, so it can store far more energy per weight than its high permeability cousin.
As long as you don't mind the transformer having a very low power throughput for its weight, then you can use a current transformer as a flyback, or any transformer for that matter, as long as the insulation is OK for your purposes. This may be OK for a quick 'proof of principle' lash-up on the bench, but for a proper product, you should really use a proper flyback, it will be cheaper, lighter, and smaller.
Keep the primary current below saturation, there's nothing to be gained and much to be lost trying to run it into saturation.