Features that replace the modifier with another use the word “instead”.
For example, the Hexblade’s Hex Warrior feature is very unambiguous:
When you attack with that weapon, you can use your Charisma modifier, instead of Strength or Dexterity, for the attack and damage rolls.
For a magic item, consider the Snicker-Snack:
While attuned to Snicker-Snack, you have proficiency with greatswords, and you can use your Charisma modifier instead of your Strength modifier for attack and damage rolls made with the weapon.
The druid cantrip shillelagh reads:
you can use your spellcasting ability instead of Strength for the attack and damage rolls of melee attacks using that weapon
All of these features make it abundantly clear that the original modifier used is replaced with a different one. The lack of language indicating replacement, such as the word instead, in the description of the Bloodshed Blade indicates that its damage is the usual damage plus your constitution modifier, 1dX + Mod + ConMod.