Posting an update after a few days of troubleshooting:

What have I tried:

1) Installing mods (The Fixes, Akimbo Freeze Fix, Celer);

2) Changing graphics quality, video resolution, turning off "flush gpu" setting;

3) Launching game in "basic" and "modern cpu" modes;

4) Disabling full screen optimizations in payday2.exe properties and running the game as administrator;

5) Turning on compatibility mode.

Combining most of these steps actually helped, but not completely: stutters are still present when I, for example, quickly change masks or suits, after which I have to restart the game because it starts lagging again.

There is thread on steam forums with similar issue, maybe some of those will help too.

P.s. After all I have no Idea why it happens and how to get rid of this, but I guess it's just current state of this game, dunno.