Keep in mind that 4KW solar may not be able to fully recharge a Powerwall in a typical Winter day even if you have no other loads running. The consequence of a drained Powerwall is more serious than some other home battery systems because it doesn't allow charging from a generator. Also, unlike most other home battery systems, Powerwall doesn't allow owners to turn on charging from the grid at will.
I don't know if this helps but here is my story:
I live in an inland valley in Southern California, which I would guess has similar sunlight to the OP's Houston. I have 1 Powerwall 2 running on a 4.5kw array (peaks at 3.6kw). I keep 15% in reserve for morning power outages and on a normal day it recharges by noon, unless we have hot and cloudy weather. I don't think I've ever had it fail to fully charge on any day. In the winter I only use about 50% power before the peak period ends (9pm) and I go back to grid use. In the winter it frequently charges well before noon. It is important to point out that I have gas appliances. My family of 4 in a single story 2500 square foot home (60's) about breaks even on electric bills with responsible air conditioning use.
I have the critical loads plus master bed/bath and home office on the battery which is why we decided to get it. It is definitely more cost effective for a grid tied system to just add more panels. We have TOU here with 3 use tiers, peak, off-peak and super-off-peak. I like the backup but am not totally sold that Tesla was the best choice long term. It is definitely a slick system but I have just about zero ability to control or modify the system. I think it is best for people who don't like to do anything for themselves. I have had a problem for the last 6 weeks where on Fridays and Saturdays the Powerwall begins to power the home early, sometimes all day on and off. It isn't supposed to do that. I've been fighting Tesla support who initially said "well the 'algorithm' sometimes does things that don't make sense to us (mere humans)." I asked her how does it make sense to burn off peak storage and run out before peak ends? After 3 weeks I got to second level support who said they would put it in a queue to evaluate and call me back next week. I called 2 weeks later and they said "yeah, something isn't right" we're working on it. That was a week ago and next week I'm calling back expecting some answers. At this point I'd be happy to find out some way to jailbreak this thing and configure it myself.
In hind sight, because I am somewhat of a tinkerer, I would have been happier with a system that I could configure and expand on myself. I was also not happy with the contractor I chose, but that is a different story. I really wish I had at least done the general contracting portion myself and subbed out the work I didn't want to do (service panel and roof work). I don't know how a generator fits into the Tesla ecosystem, but I wish I had asked. I have a total of 10kw of various generators at my house so it would be cool if I could boost this thing in a prolonged outage.
