This has a name
The general topology you are describing with a battery-powered inverter (represented by the double-conversion UPS in your diagram) forming a grid for one or more grid-tied solar inverters is called AC coupling in the solar business, and is used in some multimode (off-grid/grid-tied hybrid) and even perhaps a few off-grid systems. While theoretically feasible, and practical with the correct hardware (vs. what you have), there are some drawbacks to this setup:
It requires a rather hefty battery-inverter or premade Energy Storage System (ESS) so that the battery/ESS inverter can maintain control of your tiny "grid" when the utility supply is down and your solar generation is peaking. In particular, you need at least as many VA of battery/ESS inverter output as you have of peak solar generation capacity; as it stands right now, you're nowhere close, and without it, everything will collapse if insufficient load is present as the AC 'grid' voltage then soars, causing all the solar inverters to trip out on overvoltage and then potentially overloading the battery inverter, or causing it to shut off due to the overvoltage condition as well.
Charge control requires more care and attention in AC-coupled setups too, since the battery/ESS inverter has to be able to tell the solar inverters "whoa dude, that's too much generation!" when neither loads nor battery capacity can absorb the solar output. Most setups use what's called frequency-watt control for this purpose, where the battery/ESS inverter raises the mains frequency to tell the solar inverters to back off on the juice; you may also see references to volt-watt control as well.
Finally, as with any multimode-type solar setup, you need something smarter than your average transfer switch, or even your double-conversion UPS, to isolate your 'grid' from the utility grid when the utility grid goes dark while still permitting your solar system to export power when the grid is available. Note that this may be built into a battery inverter (along with auxiliary transfer functionality), or a separate "add-on" module such as the Backup Gateway used with the Gen2 and newer Tesla Powerwalls.
However, even with these drawbacks, it can work well enough with existing grid-tied hardware that's new enough to support niceties like frequency-watt control, and also has the advantage that there's no nasty high voltage DC roaming around the system, which eliminates the need for PV-specific arc and ground fault protection. It also permits the use of microinverters, with their attendant module-level array-optimization and rapid-shutdown advantages.
What you'd need if you wanted to do this for real
If you actually wanted an AC-coupled backup setup, you'd need to first decide if you're going to go with an integrated ESS, or an "open architecture" setup using a field-wired battery-bank connected to a multimode inverter. Current generation Powerwalls are the prime example of an integrated ESS; with their compact configuration, high degree of integration, and high power and energy densities, they are an attractive option if equipment space is at a premium or in an environment where field-integrated equipment is frowned upon. However, while the Tesla setups are capable of multimode operation using the Backup Gateway, and some other manufacturers (such as Sonnen) now support similar features, some integrated ESS systems do not provide for this configuration (look for a "backup" or similar mode in the user documentation).
The alternative option is to use what's basically an open architecture system, with a separate battery-inverter rated to UL 1741 in North America or the equivalent local grid code for where you're at in the world, DC-rated switchgear (which while non-trivial to find, is not completely unavailable), individual battery packs/blocks, and a cabinet or rack for said batteries. These systems take up much more space and require more integration work from the installer, as well as a much larger balance of system with fat cabling and high amperage, DC-rated, yet mains-circuit-suitable switching and fusing as they run at 48V DC, but provide a higher degree of customizability, and can work with inexpensive (and highly recyclable!) deep-cycle lead-acid batteries.
