If you are concerned with stability, and if you are determined not to use a generator (I'd go for a 4-stroke petrol generator - quiet, and a KW is easy)...
I'd consider running all the equipment off the inverter all the time, with an adequate charger to "float" the batteries while the mains supply is up. The power needed is almost the same, and when the mains drop, or hiccup, or fail, the equipment carries on running as if nothing had happened.
If you have occasional heavy power users - eg the laser printer - then feed them direct from the mains and don't use them during a power failure. You can always plug them into your battery supply if needed for a job.
Your two 102AH batteries will run your desktop, plus monitor, plus router and hub for about 2¼ to (max) 3 hours before they'll be exhausted - assuming your equipment needs about 400-500 watts total. That's around 2 amps AC at 230 volts.
A 4-stroke generator outside the house will run for many hours on one tankful - but of course the equipment will stop while you fire it up.
The ultimate, to me, would be the battery plus inverter "always on" so you have completely uninterrupted power, charged from the mains while the mains are "up", then a 4-stroke generator to supply the "recharge" power to the batteries when the mains are down. That way, you have the best of all worlds.