AOA,
In push button operated 4WD systems, transfer case actuator motor may be jammed, stuck or faulty, in such a problem...because when 4wd controller gives command to transfer case actuator motor to engage in low range, it cant move gears to required position hence fail to engage 4L.
i suggest remove transfer case actuator motor (pic given below) from transfer case and get it serviced..hopefully it will resolve your issue.