Leaning back slightly might help with feeling the back more prominently, if that's allowed.
I would say lots of subbmaximal sets, like 10 sets of 20 seconds once to twice a day for a week, then up the hold time to 25 seconds, or increase the sets, or decrease the rest. If performance needs to improve quickly, dedicate a lot of training energy towards it.
Also, practicing more difficult variations could make the test itself feel much easier. Maybe fat grips, hanging from towels, doing some chin ups before the hold... Or anything else to increase the difficulty a bit.