I decided to start from the simplest part: replacing Wire library to something else. The first third-party library I’ve tried was found here: http://dsscircuits.com/articles/arduino-i2c-master-library.html
Code changes were insignificant, example sketch was enough to understand, how’s library working.
And that was enough! I can now run measurements with any sampling rate, and I didn’t notice any freezing or something like that. All you need is to put default Wire library into trash can and find something more acceptable.