First make sure you have proper contract with grid/energy company for export; selling them your excess energy. You must have done that, right? Sure at that point someone also checked that your utility meter is capable of metering the export, right?
If that is taken care of, then make sure it works: get hourly metering data from the energy/grid company (sure it must be available online to you in year 2024 in Austria, yes?)
This must mean two numbers, export and import. Compare both to what the inverter application says, to rule out energy metering error.
Specifically check if there was any sunny day when you remember that battery was already showing 100% at some point in afternoon. Then there should be export. Verify that the export is recorded correctly both in the inverter and in the grid/energy company operator data.
Batteries are nasty in that they totally destroy your energy consumption if
anything goes wrong with metering. In other words, the energy flow (direction and magnitude of power flow) the inverter sees
MUST match what is measured by utility meter, pretty closely - especially close to zero.
And metering is not made easy by e.g. Chinese inverters having an overcomplicated system where a separate box is wired with RS485, then to that box is connected all three phase voltages and three CTs and the order of the phases must match as well as direction of CTs. Too many possibilities for installers to make a mistake, and mistake is hard to see, there is no intelligent diagnosis.
It is also important to know what is the
netting period your grid company uses. This varies from milliseconds to a year depending on where you live and what kind of contract you have! And while of paramount importance, it can be nearly impossible to
know what it is, you need insider information. For example, I
know this is 15 minutes in most of Finland now, but I have no idea how I would Google that or where I would get that information as a consumer. Inverter net-zeroing feedback loop frequency response is somewhere around 0.1 - 0.5 Hz, so for netting periods shorter than a few minutes, extra cost is caused by timing mismatch. Also if phases are not netted together in a 3-phase system for billing (which was the case here still a few years ago!), it will totally ruin everything (inverter keeps total power at zero, which means some phases export while others import; if you are billed for the phases separately, you pay even at zero power).
Also see
https://www.eevblog.com/forum/renewable-energy/low-power-efficiency-regulation-of-battery-hybrid-inverters-(eu-and-otherwise)/ - increased power consumption is to be expected.
I would suggest not to get grid-tied battery inverter systems without third party management system (aware of hourly energy prices; maybe participating in reserve markets). For PV timeshift only, it is not worth the investment, and extra power draw negates large part of the effect.