My reply to your duplicate thread:
I have 3 of these brand new batteries that I got my RV that I never used. Got lithium right away before my first trip. I have solar at home and I am looking to use them to mitigate time of use electricity rates. I was thinking about charging them in the middle of the night when it is 50%...
diysolarforum.com
If you did buy one more battery, 100 Ah 48V could be suitable to feed one Sunny island (available on eBay often < $2000.)
SI has 25W no-load consumption, 4W or 6W sleep mode depending on model. For one 24 hour day of grid down, that would use 600Wh out of 4800 Wh capacity. If you tell it to periodically wake up and look for loads, consumption slashed further.
You could add DC coupled PV (and a battery shunt so SI knows about the charge current) for extended grid-down use.
There are other brand inverters that would work too. Victron has a range of wattage and voltage. And other brands, but look for low no-load consumption as Sunshine says.
Lithium has its place as well, especially daily cycling including time shifting you thought of. AGM is robust, reliable, simple, but better for standby to use when power fails.
Lithium can now be purchased for the same cash price as AGM so can be quite attractive. But it does have its own quirks, and needs to be large enough for the application. AGM will act like an automotive starting battery, very high loads for a short time can be supported, e.g. 11kW surge from inverter, using 100 Ah battery.