Pasito

Seems you are mixing options here.
Splitting:
- Do you have enough space in the DB or do you need to add another DB
- Can you do this yourself and issue COC or will you need someone for the COC
- It requires a fair amount of extra breakers and wiring and work even if you do it yourself
- The backup breakers are feeding from the inverter so always on (no switchover)
Switchover:
- Manual.  It is an either/or situation but not both grid power and backup power.
- Not sure if, or how an automatic system will work.
- You have a delay or short interruption in power.
- Don't know of anyone using such an option
Direct or everything on:
- Since everything runs through the inverter your inverter needs to be big enough for the highest load.
- For me with geyser and stove etc that would be 8kW and up. My neighbour does this with two axpert inverters
- The saving on not splitting your DB can go to a larger inverter.
PS Larger inverters normally use 48V batteries while some of the 3kW inverters can be 24V thus saving on batteries if you just want to bridge loadshedding.
Maybe an additional small DB for the backup circuits with a 3kW inverter is the cheapest option for you.
Another option is to install the Inverter close to the load or equipment to be on backup and have an outlet at the inverter using leads to all your equipment. You will then need some floor standing lights or something.

I recently found out that the sunsynk 3.6kw is fanless, exclusive to SA model, but it is the same price as the 5kw with fan 🤨

The important thing is that you know your loads and the maximum you will need at any given time. I think this is important.

You need completely silent inverter. Should be fanless model - check the specs in the datasheet for whatever power you think, you will require (for your needs maybe 2kW will be more than enough).
It would be best, if it supports LIFEPO4 batteries, so buy 2 x 12V LIFEPO4 and a trolley enclosure for them and put the inverter on top.