The datasheet tells you that.What is the minimum/maximum V the AD623 can handle on the inputs?
Unless you need to offset your output just use GND (as Eric suggested).I did see some designs that use a ref voltage IC (i.e.: https://www.ti.com/product/ref1004-2.5 ) - is that something I should try?
If I understand the datasheet correctly any interference on the inputs can get rectified and appear as an output error. That may be why your output is stuck at -2.7V.I am not too concerned by the 50Hz interference. Will be hard to avoid on a breadboard design I think.