In addition to the scope, many clamp-on ammeters have the ability to measure inrush current. As long as you understand that they are sampled measurements and can miss the true peak current, they may get you close to want you want. If you really need to know the actual peak current, use a scope with a suitable transducer (current transformer, clamp-on current probe, Hall effect probe, resistor, etc.).
The min-max hold on most meters requires a certain number of samples to register - and may not be accurate for in-rush capture. Eg the Fluke 87 requires about 100ms I believe and then does an average.
A current probe and an oscilloscope will work. Tektronix makes a couple of
different probes and interface modules (amplifiers). The amplifier output plugs
into the oscilloscope input.