Answering your questions in turn:
The second battery is daisy chained into the first battery and your inverter will see it as one giant battery. In practice both batteries will charge and discharge together but the first battery in the chain will probably charge and discharge a bit faster than than the second, and so the SoC's of the two batteries will drift apart, typically 10-20%.
It is important to be aware that when one battery reaches 100% or 4% then (full or empty) then all batteries will stop charging/discharging. Therefore you won't get access to the full 9.5 capacity, you'll probably get up to 20% less. This is annoying but lots of people have confirmed that that's what happens to them.
Adding a second battery won't change the throughput of your inverter so you won't be able to charge or discharge the battery any faster, just you'll have more capacity. So if you charge overnight you'll be able to charge for longer but you'll have to do the maths to work out whether you can completely fill the battery in your cheap rate period.
Whether this makes financial sense or not, hard to say. You have said that your current battery is sufficient in summer so the only benefit of additional storage in summer is that you can charge the batteries more and then if you have a peak-rate export tariff like Octopus Flux, you can discharge more in the peak and still give yourself enough to run the house through the evening.
Winter it comes down to your overnight charge period and whether you can get enough charging in that the overnight vs day rate saving makes it worthwhile.
I have two inverters and one battery on each so with two inverters I get double the charge and discharge rate. I've looked several times at getting a 3rd battery but concluded every time that financially its just not worth it. Depending on the scenarios I took and assuming rates remained flat, the payback for a 3rd battery was between 6-22 years. of course economics are not everything but its important.
And your last Q about who installs it. Any GivEnergy supplier can install an additional battery for you but they have to take ownership of the entire system if they do so. There's no concept of different bits being supported by different companies