I cannot speak for the OP's equipment, but my own experience with Apple is that keeping the AC power on when using the iPad does not seem to compromise the battery, and in fact makes it lasts a lot longer if I ever do then take it off charge, than if I unplug it as soon as it reports 100% full.
This iPad is quite old now, and it started "dying early", when the battery was still showing 20-26% capacity. I solved this problem by using a small capacity charger rather than the large capacity one, and the same iPad with the same usage pattern, now lasts down until the battery shows 1% charge.
I think the reason is that the internal resistance of the battery gradually increases as it ages, and if a high capacity charger is used the higher internal resistance can make the battery charge state appear higher than it really is; at which point the iPad charging monitor switches off the charging. This actually leaves the battery only, say, 75% charged, but I found that using the smaller capacity charger - slower for longer - gets the battery much closer to a full charge. Then leaving the charger connected when at my desk gets it even closer to properly full.
Concerns about unattended lithium chargers could be valid. I have never had an issue, but I only use genuine Apple chargers and gear.
This battery must have been through well over 1,000 cycles, and although it will only last a couple of hours on continuous battery-only use, it is still soldiering on very well.