The E023 error code is an overload warning that the load exceeds the output limits. Most likely when the compressor starts it momentarily exceeds the 9 amp maximum output limit of the DC socket. When compressors start they require a larger amount of electricity than when running. Usually this is for a very short period of time and the load demand quickly goes back to normal. Your Bluetti is recognizing this overload. The hotter the outside and inside the cooler and the colder the thermostat is set, the harder the compressor has to work to reach the desired temp.
Yes, dropping the first bar in about 4 hours sounds normal. The battery bar is an approximation of remaining capacity and since the bars represent a range of + or - 20% at a given time it should be viewed as an approximation and not a specific amount of battery life left.
Here is a link to the operators manual that is available through the Bluetti web site.
The manual is full of useful information and lists all the error or warning codes along with their meanings. Your previous question about the 017 error code was stating that the solar panel combination you had connected exceeded the maximum allowable solar charging input voltage. Too high a voltage can damage the machine as well.