Since it happens after things have been idle for a while, the host is probably trying to power down the Quadstick due to inactivity. In Windows there is a setting to prevent this but I don't know how you would do this with an iPad. Maybe inserting a powered USB hub in-between the iPad and Quadstick would prevent it.
Another workaround might be to keep the Quadstick "active" by having it constantly change some output value repeatedly so the iPad never goes to sleep.
A configuration file with a line like this would do it:
| x | pulse 10000 10000 | mp_left_sip | | | | | | | | press and release the cross button every ten seconds for 2 days |
The output, in this case the Playstation Cross button, would turn on and off slowly for a couple of days. Switching away from whatever mode or file contained that row would clear the output.
Row 16.
For the iPad, maybe a mouse button, letter, or arrow key, would be a good output to use. A game controller button might work well. It only needs to keep tickling the iPad, not actually do anything.
When you are done using the Quadstick, switch to a mode that has that row and trigger it and it will keep pulsing for a couple of days, or until you switch to a regular mode.