Beta is nothing to do with this, beta is the small signal current gain in linear operation, which does not
apply to switching (where the transistor is either saturated or cut-off, not in the forward region).
In saturation you need the base current to be about 5 to 10% of the collector current for a BJT.
For that amount of stall current a MOSFET or darlington are needed, a single BJT cannot do the job.