Minimum Current(not for switching) On a Relay?
Looking at the specs printed on top of your relay it seems like it has a DC rating on it.  So the relay alone is probably fine for switching DC.

Note that relays have different ratings for DC because there is no zero crossing in the current to extinguish any arcing.  This can cause extra wear on the contacts or even cause them to weld shut if used outside the DC rating.

There's also often a minimum current required to keep the contacts wetted and break through any oxide layer that may form.  Switching currents below that can cause high resistance contact unless the relay is rated for "dry" contact operation.

It's possible that for your device there are other reasons why it will only work with AC mains voltage.  It may be that they use some of the AC voltage to feed other parts of the circuit (even though you have USB power) or there may be some type of safety function built in that only works with AC.  There's no way to tell without knowing the details of the design.