Motor starting surge is about 5x operating current, and only lasts a fraction of a second. Too long for most capacitors to support, however.
A very small battery, but capable of high current, should let a system start then run much larger loads.
If lead-acid, you would want low battery cutout to be quite a high voltage, keeping SoC above 80% or 90%, so it can give 10 to 20 year life. Voltage will drop low at high current, so inverter needs to allow some seconds before shutting down if it doesn't understand Peukert.
Possibly super-capacitor bank would also work. But I think used automotive batteries from the wrecking yard would be an economical first implementation. Handy as spare or jump starter for your vehicles too.
One link somewhere around here had video of a contractor who installed 6kW Sunny Boys for a customer, and they demonstrated that Secure Power was able to start a large angle grinder. That looked like one that might draw 15A full load, so I think Secure Power may deliver surge well over the 2000W rating they quote. System is sized for 240Vrms output, so when buck converter is delivering 120Vrms the capacitor voltage can swing a lot to release stored energy.
I think solar with large loads in the daytime is a reasonable goal. I run my central A/C with a fraction of PV output.
DC or AC powered LED lights and communications could be powered 24/7 from battery. Much of the gear is 12V, so separate AGM battery and charger could be used.