It's almost certainly not impossible to do this in sed, but also almost certainly somewhere between hard and impossible to understand the solution some weeks from now unless you are extremely familiar with sed. Better then to switch to a more human-readable scripting language:

awk '/^BEGIN$/ && !s { s=2 }
!s
/^BREAKPOINT$/ && s { s-- }' file

In very brief, the variable s keeps track of whether we have seen the beginning separator and if so how many times we want to see the terminating separator before reverting to printing the input lines.

With your shown samples only, please try following awk code, written and tested in GNU awk.

awk '
/^BEGIN/,/^BREAKPOINT/{
  flag=1
  next
}
flag && /BREAKPOINT/{
  flag=""
  next
}
!flag
' Input_file

Explanation: Adding detailed explanation for above.

awk '                     ##Starting awk program from here.
/^BEGIN/,/^BREAKPOINT/{   ##Checking range from line starts from BEGIN till line starts BREAKPOINT then do following.
  flag=1                  ##Setting flag to 1 here.
  next                    ##next will skip all further statements from here.
}
flag && /BREAKPOINT/{     ##Checking condition if flag is SET and BREAKPOINT is found then do following.
  flag=""                 ##Nullifying flag here.
  next                    ##next will skip all further statements from here.
}
!flag                     ##Checking condition if flag is NULL then print current line.
' Input_file              ##Mentioning Input_file name here.

A perl:

perl -0777 -pe 's/^BEGIN[\s\S]*?^BREAKPOINT[\s\S]*?^BREAKPOINT\R//gm' file

Or,

perl -0777 -pe 's/^BEGIN(?:[\s\S]*?^BREAKPOINT\R){2}//gm' file