My first thought is that you need more of both PV output, and battery capacity.
There are just the two of us at home, and we are fairly careful, but we do have oddities such as a borehole pump for our water as extra, and cooking has to be electric. We have the usual 4kWp system, and only a few months back, had a 5.2 kWh battery installed. this time of year the battery goes totally flat in the evening, and its just staggered up to 50% charge today (yesterday it barely exceeded 5%). In the summer this heats all our water, and fully charges the battery, which will then only go down to maybe 15% by the morning.
I'm not sure if it has changed sine the demise of the Feed in tariff, but the effective maximum on the usual connection protocol was claimed to be 4kW, but it wasn't, that's a porkie, it was only 16 amps, so actually 3.68 kW. if you can't fit more than 3.4kW worth up there, then there is no discussion, but I'd aim for as much as you can manage, especially as the efficiency of panels has gone up a bit since ours were installed 5 years ago, so you can squeeze a bit more in than previously. There is logic in having panels capable of generating more at peak ties than the inverter can export, as it means that more of the time you are generating more, as most of the time you are at nowhere near peak output.