Electrical – Photovoltaic installation: does it make sense (or is possible at all) to get a bigger inverter than the power output of the installed panels

You need an on-grid or grid-tie inverter, and you can buy them far cheaper than you can possibly build them. In any case the power company and electricity regulator won't let you implement the circuits yourself unless you're prepared to spend thousands on certification tests, including requirements to, for example, shutdown within a millisecond of mains loss.

If you're in the northern hemisphere, the lower angle of the sun in the winter is probably contributing to your low numbers. (When the sun is low in the sky it passes through more atmosphere to reach the ground.) Presumably you'll do better in the summer.

Also, the tilt angle of the panels should be matched to your latitude. If your panels are mounted on your roof and aren't at the optimum angle (or aren't pointed directly south), that would decrease your efficiency. In any event, unless you have tracking panels or constantly change the tilt, your panels will only be pointed exactly at the sun twice a year, and at all other times will be slightly misaligned.

This website has a detailed description of tilt angles, but the net result is that if you have a fixed array at the optimum angle, you'll only be getting 71% of the exposure than if you had a tracking array constantly pointed at the sun. If your angle is not optimum you will do even worse, obviously.

EDIT: I should also point out that your definition of "optimum angle" for a fixed array is dependent on what you're trying to optimize. There are a couple of ways to look at it:

1. Generate as much power as possible over the entire year. The 71% refers to this option.
2. Generate as much power as possible in the summer, when electricity usage may be higher due to AC and/or power may cost more to buy from the grid. This will be a lower angle [array pointed more upwards] than 1), since the sun is higher in the sky in the summer. Winter usage will be lower than 1).
3. Try to generate power evenly throughout the year. This will be a larger angle [array pointed more towards the horizon]. This approach tries to increase power generation in the winter relative to 1), at the expense of some summer power.