I'm surprised this makes sense. When I priced out solar panels and batteries for a similar setup, earlier this year, it was going to take decades just to break even.
Did you look into 12v mini split systems? Avoiding the need for an inverter might drastically bring down costs and efficiency. I’ve seen a couple sets out on the market that come with everything already (prefilled mini split system and solar panels). Adding a transfer switch and an inverter could get you somewhere! I liked the idea of it purely as an “assist” with a larger system. Pull some heat out of the house to reduce the load on the main hvac system.