Just a small reminder that this answer is time limited in relevance: for current travel it is only very limited in possible application - don't take this as advice that it is possible to do this all the time without visa, you'll get a lot of problems.
Even though there are negotiations about implementing mutual visa-recognition between the Union State members Russia and Belarus, at this time it is not something you can rely on.
For your plan in June, you can absolutely do this visa-free if you
change the visa for a ticket to the European Games, as the organizing
committee was able to confirm to me (I'm flying from Kaliningrad).
You just need to have the medical insurance you'd need anyway,
possibly register the ticket for data sharing with Russia (it can't be
wrong), and you can only use your ticket to travel to Belarus once,
which must be before the end of the games. Afterwards you can still
use it to transit to another location via Russia.
See my answer to your other question.