The bec is the morningstar of polearms imo it doesn't need any buffs since it's longer, can thrust and has 1 more speed of morningstar but not unbalanced.
Maybe lower it's price to 5.5k?
Let's not compare shoes and potatoes
the animation are not the same thus reach and speed also.
The issue here is that it is a no brainer to take LHSP rather than bec de corbin in the polearm category.
Howerver with three heirloom each the two weapon seems balanced
longer reach + knockdown against better speed and damage
bec de corbin
weight 2.8
requirement 14
spd rtng 93
weapon length 120
swing damage 36, pierce
thrust damage 26 pierce
Gen1= 94spd 38pierce 29pierce
Gen2= 94spd 40pierce 29pierce
Gen3= 95spd 42pierce 30pierce
LHSM
weight 3
requirement 13
spd rtng 94
weapon length 138
swing damage 34, blunt
thrust damage 20 blunt
Knockdown
Gen1= 94spd 37blunt 22blunt
Gen2= 94spd 39blunt 23blunt
Gen3= 94spd 40blunt 24blunt