NO what will happen the charger will try to feed your inverter + the amps the inverter is pulling.
A battery /battery bank is more or less a sponge that lets it's self be filled with --------fare less demand---- and the charger is made for that in mind.
I know than you say why can the MPPT charge controller do it with the solar panels and again it was designed for charging and using your inverter at the same time.
One way to get to where you would like to be is if where you are in the world if you can find used golfcart batteries.
When I first was figuring all this out I bought 10 6volt used batteries and wired two in series to make 12 volt sources than wired those 5 sets in parallel for my 12volt bank.
I use AGM batteries now but cost of those was a leap.
The only way I could see you right now powering system is with a larger battery bank, more solar panels and or a wind generator when there is no sun that might buy you more run time at night with less batteries.
Or just realize you might have down time to top off system and only be able to use than.
Don't freak about the price I only give you this link for educational purposes to understand what I meant by wind generator.
Hybrid Solar and Wind kits are the easy way to bring clean power to your home no matter your experience. Get the equipment you need to break free of the grid.
windandsolar.com
But again to help your with your question about using the smart battery charger and using the inverter at the same time your going to have magic smoke or fire.