Stew

Based on the specs @Moffat posted a 1600W will either have 6 or 8 batteries in series and yours only 4.
With these series connections I would rather not use that number of lithium batteries as the different batteries might be running at different voltages and each internal BMS would just cater for it's own cells. Also costly to try it by trail and error. 

If you have a manual with your UPS just check if you can change the charging parameters of the "new lithium batteries", if not you may end up being stuck with having to continue using AGM batteries as the lithiums will continuously be undercharged. They may last longer than the AGM for a little while but because they are in an enclosure you'll not notice that they have swollen. I've had a bad experience with Bluenova batteries and the warranty was voided because they were continuously undercharged and ended up swelling and retailer refused to honour warranty even though it was touted as a direct drop-in replacement for my 12v 7.2Ah battery. Ensure your inverter can charge up to 13v as in the Bluenova or whatever brand you choose otherwise you may go through the same bad experience I went through. Over-voltage side is not too much of a concern as the BMS will ensure this protects the battery, but there's no under-voltage protection as this is limited by your UPS.
 

You're welcome and best wishes and keep us posted.

If money is tight get a trolley inverter and run a few extension cords
Dont consider anything other than lithium as my sole experience in gel batteries resulted in a total write off after 90 days

Where abouts do you live? In case somebody wants to refer you to a local business.

RE your financial situation. The cheap way to get going here is some kind of lead/acid batteries. But with all the loadshedding we have, those are going to take a pounding and may not last long. Lithiums will cost you significantly more to get going, but they will last a lot longer, can be discharged to a greater depth, more regularly, and will be a better long term proposition.
Yes. You can get an electrician to do that. It means changes to the DB thus a new COC will be required.
You've made a good start. Two pieces of information are required to get started, and I think you have or are close to the first
1) Consumption per hour
2) Peak momentary consumption across the backed up circuits. Thinking of your fridges and freezer, they are likely to draw more very briefly when the motor starts, then settle down. But it only needs a momentary overload to trip an inverter.