I’d point out that the “Current - Peak output” parameter in the PMIC - Gate Drivers family refers to the output capability of the IC in question, rather than the resulting capability of the circuit in which it is used. Whatever transistors you hang off the outputs will increase the output current capability of the driver.
Toward that end, the IRS2186 looks like it could be a reasonable choice. Of the two output devices you mention having, the 600V IGBTs are the clear choice over the 150V FETs, for the simple reason that the latter offers no margin between your desired output voltage and the rated smoke voltage of the part.
Generally speaking, IGBTs get used for high-voltage/lower frequency switching, whereas FETs do better at higher frequencies and lower voltages. The line of demarcation shifts with time & technological development and I’d suspect that a 150V/20kHz switching application might favor a FET at this point, but the details of your application might skew that tradeoff.
In any event, you’ll want to be very mindful of parasitic inductances in your physical implementation–they’ll slow your rise times and can make unexpectedly large voltages, which is an easy way to have things blow up on you. Particularly with your load current and gate drive traces, you’ll want to minimize the area encircled by current flowing in a circuit, by keeping them short and minimizing the distance between the outbound and inbound conductors. Routing everything on the top layer of a PCB with a common, contiguous, unmolested ground plane beneath is a good start to achieving this. A ratsnest of wires flying here & there on a solderless breadboard, not so much.
Something else to note is that most of these half-bridge gate driver ICs don’t do well operating in a continuous output-high mode, due to eventual loss of charge in the bootstrap capacitor. But maybe that’s not an issue for you–again, lots of things hide in the details. Any additional info or color commentary you can provide on your application/ needs would be useful in offering you more specific guidance.