Well the thing I know about Inverter to UPS is that the UPS will work fine if the Inverter is pure sine wave. If its not, then the UPS will think that the power supplied is bad for the electrical devices connected to it and will switch to the backup battery to protect them.I have used a UPS on an inverter line. I have used this UPS to connect to a desktop system. Is it safe for the UPS since it is on an inverter line. Earlier I had used the desktop system in the inverter line itself but due to fluctuation the system used to reboot once in a while. I have connected the UPS now to the inverter line and the system does not reboot now when there is fluctuation but not sure if it is safe to connect the UPS to inverter line and will that drain out the battery in the UPS soon.
If the inverter is pure sine wave and you're just using the UPS as a middle man to handle the voltage fluctuations then you should be totally fine.
But then why are you worried about the battery life of the UPS when you're already using the inverter for the backup whenever electricity gets cut off?? The UPS will most likely never even switch to its backup batteries as the inverter will provide it continuous power during power cuts