A more careful reading of the manual might have helped you.. page 32, where the upper knobs are explained:
'1ms PEAK Button: Used to select the 1ms peak capture mode (VAC only)..'
So it's not working in any other mode, especially NOT in DC current mode.
Your 150ms long DC pulse @ 50% D.C. will be averaged by any of your DMMs, but the different integration methods of their individual A/D converters will give different results, and their readings will also vary from sample to sample, depending on the timing of conversion vs. your pulse duration.
None of them will measure the peaking current, I assume. In fact, you really don't know WHAT the other, supposed 'better' DMMs will measure. It's only random stuff, supposedly.
So you have to chose another method, maybe a bench DMM with fast DCI digitizing possibility, where you can scroll through the acquired data, and measuring peak current by means of cursors. KS 34465A can do so, for example.
Frank