Don't blindly trust one DMM.
This is a key point. If I had (besides budget) more room, I would get a Keysight 34465 bench meter and a good handheld DMM, say Fluke 287/289.
(Btw, the Flukes are here in Europe twice the price of the US market prices.)
I am interested in low voltage AC (down to a few mV) for some audio connected measurements.
Ideally, I would find a handheld DMM with the accuracy of the 34465, but this does not exist.
I would also like to simply trust the product specifications to avoid having several quality devices that can measure (by specification) quite accurately down to 1 mV AC.
Not sure, if it is smart to fully trust the specifications. Maybe this is indeed blind trust.
The inability of the 34401A and the 287 to display very low mV AC values surprised me a bit.
Actually, the 287 does not meet its specifications with your observation that 0.150 mV 1 kHz is displayed as 0.
The accuracy as stated by Fluke should be 0.4% + 45 digits (0.001 mV) in this case.