cecilh

Yes. But I should not have to do that. Looks like Synsyk is cutting corners Saving R80 on a R25k plus inverter seems pretty short sighted.

Excess will be sourced from Eskom. Non-essentials generally use eskom unless you setup your CT and settings to export solar to them.
Inverter will trip if load on essentials is greater than 5kw during LS or greater than 7.5kw when there is no LS (35A pass through)

If wired correctly, your non-essentials remain on your 'main' DB board and hence are wired before your inverter. Any power draw from them does not go through your inverter - which means does not impact the 5kW that your inverter can provide on the output side. Your essentials are connected to the output side of the inverter, which are restricted to 5kW of loads during loadshedding and have a 35A pass-through capability for access loads while the grid is present. I have to admit that I do not know though if this 35A pass-through is total load (as @mzezman wrote) or in addition to the 5kW inverter capacity. Maybe @Leshen can advise please?
The beauty of the Sunsynk is that it is a bi-directional inverter - which means it can not only draw power through the AC Input, but also push back power through it. This means, that (if your CT coil is correctly installed) when it has access PV available (meaning when your available PV is higher than your essential load) it can push power back to your non-essentials. Hence your non-essentials can also be supplied by PV, reducing your grid draw. If your non-essential load is higher than your excess PV, it will take the balance from the grid. However, this will only work while the grid is present - so not during loadshedding.
Assuming that your essential load does not make up more than 5kW of your 7kW load (not kWh - different thing 😉), it will most definitely not trip your inverter but rather:
If your inverter settings do allow you to draw from batteries (SOC higher than target SOC):
It will provide a total of 5kW from your PV and batteries to both your essentials and non-essentials and take the 2kW shortfall for the non-essentials from grid If your inverter settings do not allow you to draw from batteries (SOC lower than target SOC):
It will provide whatever load is present on your essentials and use the remaining PV to charge your batteries. Non-essentials will be completely powered by grid
(I have to admit I am not 100% sure on this. I cannot exactly recall as I no longer have anything connected to non-essentials) 

If your total load is 7kw, and your panels supply 3kw, the battery will only supply 2kw as the limitation is 5kw. The other 2kw will be supplied from Eskom. There is no limitation on the non essential circuit and there is a maximum of approximately 8kw (5kw DC to AC) (3kw from Eskom) limitation on the essential circuit when Eskom is available and 5kw when Eskom isn’t available. The issue with using more than 5kw on the essential is if Eskom goes offline at that instant, the inverter will shutdown due to AC overload fault.