Good question - essentially you're looking for a powerbank with a very low threshhold for auto-shutoff.
Just tried my Aeropex with the Aukey powerbank I had handy and it shuts off as you describe. .
Here's a thought - what if you try a two-port powerbank and plug your phone into the 2nd port? Would that keep it awake sufficiently? Or perhaps one of those USB fans? Basically something to draw enough current to keep the powerbank from shutting itself off.