Preparing for the Hunt; A Blood Hunter Handbook

[Sicarius Victis]

I would say yeah, as it's a boost to the weapons attack; if I have a barbarian or a half orc a frost band weapon, it would get extra dice for the ice damage as well. If visual ques can be reviewed, the paladin in the players handbook is a half-Orc as well

I believe you're misunderstanding the ability.
"When you score a critical hit with a melee weapon attack, you can roll one of the weapon's damage dice again and add it to the extra damage of the critical hit."
First, it explicitly states "one of the weapon's damage dice, not any other source. Second, it only allows you to add a single additional die no matter how many dice you're rolling for the critical hit. Thus, with this feature, on a crit with a greataxe they'll be doing an extra 1d12 damage, but on a crit with a greatsword they'll only be doing an extra 1d6 damage.

So no, it does not allow you to add more Crimson Rite damage beyond the standard additional for the critical hit.