Welcome, Marcin, to the OEM forum, and thank you for those kind words.
The problem with the RFM12B is it doesn’t have an internal buffer big enough to hold the complete message, it is only one byte big - therefore the JeeLib library needs to send the data one byte at a time with no gaps. And that is where it fails - the processor is too busy maintaining phase lock. What actually happens is the data has gaps - so it’s corrupted, and the sketch loses phase lock - so that’s wrong too.
The RFM69CW does have a big enough buffer, so that can easily be filled a few bytes at a time, then the RFM69CW does the rest without any help from the main processor.
You don’t need to remove the RFM12B, simply do not use it. Although you can use an ESP8266, my colleague @borpin recommends a Pi Zero W in preference. Note you can only send serial data slowly at 9600 baud or less.