Below which voltage should an AGM deep cycle battery cease to supply charge to devices
Battery aging is inevitable. It occurs during charge transfers and life cycles reduce 50% for every 10'C rise above room temp and to some extent by the minimum % DOD or cutoff voltage you choose.
You must define a spec to choose desired %DoD and thus capacity and thus lower cutoff voltage and thus lower cumulative Ah capacity* charge cycles.
As always, refer to your AGM datasheet.
This datasheet from Victron shows the the number of Charge cycle always increases by using a lower %Depth of Discharge (DoD). This implies a higher &quot;resting&quot; cutoff voltage gives longer life.  But how much?  When I multiply the # of cycles from the barchart by the amount of charge used (DoD) the total life cycle charge goes up from 300 effective total charges to 435.

( measurements taken from bargraph by counting pixels with I_view)
Conclusion: Reducing your cutoff will extend inverter runtime but also reduce battery number of days of charge life, so you lose time later.
Aging is directly related to total charge transferred during life. For longest lifespan,  keep at room temp! (20~23'C)  and choose  Mfg's advice wisely.
If you need more capacity, use more batteries rather than lower the cutoff, but if desperate, you can reduce cutoff to 10.8V for the best short term capacity.