It really depends on how many hours you expect to need the battery power - 5 hours? 1 hour? How big is the tv and is it energy efficient? Your PC desktop has a motor in it so will require more power than you would think - would be cheaper to buy a new laptop than to buy power for the desktop.
For lithium, it depends on type. If it's li-ion, stay away - way too volatile. If it's lithium Ferrous, (LiFePO) it really depends on brand, and the good ones will last 20 years. You can get really good NMC batteries (same chemistry as Tesla) for a decent price, and they will last 30 years.
3kva is probably too big a size of an inverter. You will save a lot of money if you get a smaller one. For what you are doing (no fridge or fan?), it seems that you don't have a lot of appliances with surge (appliances with motors, when plugged in can surge 2-3 times their operating wattage) so you need an inverter capable of handling that. But you're probably operating around 1kw per hour, so 1.5 -2kva inverter will do.
I found the cheapest brand for UPS grid charging (no solar) to serve as a backup, with both inverter and charging combo, with great value is Magnum - made in USA. You can also buy Outback UPS but is twice the cost as Outback.