capacitor selection for power boost
What you are suggesting makes no sense.  Capacitors don't work how you apparently think they do.  Capacitors can temporarily hold up a DC power supply, but not AC as you have.

Especially considering your apparent level of electronic understanding, the answer is to go buy a bigger inverter.  No, seriously.  Fundamentally you have a 20 pounds of stuff in a 10 pound bag problem.  Get a bigger bag, or stop trying to cram so much stuff into it.

Your numbers also don't add up.  Your inverter is supposed to be able to handle 4 kW for 10 seconds at 110 V.  That means 36 A.  You're not exceeding that.  Possibly the input to the inverter can't keep up with the load.  Is this battery really able to supply 4 kW for 10 seconds.  If this is 12 V, for example, then that would be 333 A.  Even if the battery can do that, maybe the cable between the battery and the inverter is dropping too much voltage.

Again, the basic problem is that your system is under-specified.  Either the battery, inverter, or the connections between can't handle the full power, or can't really deliver as promised by the sales BS.