How to increase output power of this simple CD4047 based inverter
Probably: replace the design with something that actually is fit to drive an inverter – there's plenty of ICs that are actually designed to drive inverters, and have things like feedback loops. Without a lot of time on your hands, and an in-depth understanding of what's happening and why, you won't be able to improve this very minimalistic "demo" circuit.
On the other hand, there's a lot of application notes from semiconductor manufacturers that actually explain the theory of operation of a device – and list the limits of a design. It'll be easier to do this well from scratch than to save this particular design.
A typical design would use a flyback controller to feed a full-bridge MOS set at more than 100 kHz, keeping the transformer small (cost reduction without reducing power).
The resulting high-frequency AC would be rectified using a classical diode bridge rectifier, generating the intermediate circuit's DC voltage.
From that DC voltage, another four-FET controller would "push" current through one or two inductors and your load, with some kind of feedback mechanism that strives to achieve a sinusoidal output, or just pre-calculated 50 Hz switching.
From TI's SPRABW0B:
It's significantly more complicated than your CD4047 inverter, but you actually get a lot more efficiency out of this – so you don't have to buy a higher-rated transformer or overdrive your output voltage to power regular loads.