I'm looking to pick a regulator for an application that will require 400 mA and lower current bursts for 300us at a time every 0.05 seconds. What do I look for in the datasheet to see if the regulator is capable of that or should I just load the output up with capacitence? For example, one such regulator I was looking at was the ISL9111AEH50Z-T7A, datasheet here:

https://www.renesas.com/us/en/document/dst/isl9111-isl9111a-datasheet

The regulator will be powering a diode for very short bursts but also be powering a microcontroller and Nand gate.

 

Very large Ceramic-Capacitor, ( maybe 2uf ), plus a large Bulk-Capacitor, will be fine.
Check your chosen Regulator's Data-Sheet for quirkiness
regarding Input and Output Capacitor requirements or problems.
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The critical parameter for your application is "response time to step load change." Adding low resistance capacitors as already suggested is the other part of what you need.