PID Control Object (with questions)

You could try calling reset(true) on the left-hand side before upload just in case any data was left in memory from previous saves.

The issue is not your code itself, but that you're trying to cram MQTT, BME680 and PID into an ESP8266.

You could use something like an Espruino WiFi that has more than enough space for this, however if you're after some ideas to save space you could ask on the ESP8266 section of the forum rather than hijacking this thread. I believe there is a smaller MQTT module available.