Very few batteries will discharge at above 3.5kW, unless you are using more than one or a Tesla PW2.
For your solar you can just use two arrays one that is feeding back the the grid at 3.68kW and then one isolated if you really want a full 10kWp system. Adds cost, but so does having a larger than normal array. It's always good having more generating capacity if it can be supported, but you need to weight that up with what the DNO is allowing you to do on your local circuit.
How much energy do you currently use overall, and how much of that is inside of sunshine hours?
The battery was either PW2 or SolarEdge 10kWh, the latter can provide continuous output at 5kW according to the datasheet. Our energy usage is fairly flat through the day, all things considered. So we'd be needing plenty of storage. The plan all along was to go big on generation and get multiple batteries but then if we have more than one, presumably only one could discharge at a time (and limited).
OK thought about it a little and realised you can probably generate more (have more panels) but have a limiter and battery setup so that the export is restricted?
As it has been in place for years I imagine it has to be so otherwise you would see less coverage on roofs and the installers would be plastering disclaimer limitations etc.
Right but limiting through the inverter would also limit power going into the home from the PV system, I think?
You can have a 10kW inverter if you wish, and just limit its abilty to export to the grid at no more then 3.68 etc. Anything generated that isn't used or stored in the battery above the 3.86 figure, would just be wasted if you had spare generation.
As above. So I think going back to Journey's post, we'd need to have an isolated system somehow?