I think @itu is right. A standard 9V battery can't supply huge amounts of current, and between any losses in the step-down circuit and the start-up draw of the pedal it probably can't cope.
You can try it from the 9V pedal supply, that ouught to be able to handle the startup current. If that doesn't work then it may be the lighter circuit is a problem. A 9V-5V converter is only a few quid on Amazon:
https://www.amazon.co.uk/Converter-Module-Transformer-Voltage-Regulator/dp/B09B829DL9
It's also designed for the job.
If none of that works then an alternative could be to use a USB power bank as the 5V supply, strapped under the pedalboard.