Can a Damphir's bite be used as a Monk's dedicated weapon?  RAW it fits the criteria...

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Dedicated Weapon

2nd-level monk feature
You train yourself to use a variety of weapons as monk weapons, not just simple melee weapons and shortswords. Whenever you finish a short or long rest, you can touch one weapon, focus your ki on it, and then count that weapon as a monk weapon until you use this feature again.

The chosen weapon must meet these criteria:

• The weapon must be a simple or martial weapon.
• You must be proficient with it.
• It must lack the heavy and special properties.

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Vampiric Bite

Your fanged bite is a natural weapon, which counts as a simple melee weapon with which you are proficient. You add your Constitution modifier, instead of your Strength modifier, to the attack and damage rolls when you attack with this bite. It deals 1d4 piercing damage on a hit. While you are missing half or more of your hit points, you have advantage on attack rolls you make with this bite.

When you attack with this bite and hit a creature that isn’t a Construct or an Undead, you can empower yourself in one of the following ways of your choice:

• You regain hit points equal to the piercing damage dealt by the bite.
• You gain a bonus to the next ability check or attack roll you make; the bonus equals the piercing damage dealt by the bite.

You can empower yourself with this bite a number of times equal to your proficiency bonus, and you regain all expended uses when you finish a long rest.

It is already a monk weapon: 

[monk weapons] are shortswords and any simple melee weapons that don’t have the two-handed or heavy property.

Unfortunately you cant use a Con based weapon as a Monk weapon. RAW for Martial Arts below

• You can use Dexterity instead of Strength for the attack and damage rolls of your unarmed strikes and monk weapons.

Hence hoping to use the Dedicated weapon option.

Quote from anywherebutiraq >>

Unfortunately you cant use a Con based weapon as a Monk weapon. RAW for Martial Arts below

• You can use Dexterity instead of Strength for the attack and damage rolls of your unarmed strikes and monk weapons.

 That is a benefit of being a monk weapon, not a requirement to be one.

If that is your hang up, making it a dedicated monk weapon doesn't change anything.

It still gets the increased damage die either way at least.

Quote from DxJxC >>

Quote from anywherebutiraq >>

Unfortunately you cant use a Con based weapon as a Monk weapon. RAW for Martial Arts below

• You can use Dexterity instead of Strength for the attack and damage rolls of your unarmed strikes and monk weapons.

 That is a benefit of being a monk weapon, not a requirement to be one.

If that is your hang up, making it a dedicated monk weapon doesn't change anything.

It still gets the increased damage die either way at least

 

To hit is more the concern that damage output.

Crowd sourcing my RAW readings. :)

I think it is probably still CON, now that you pointed it out.

The damage die definitely increases and you can stunning strike with it and use bonus action martial Arts/flurry of blows after it. But since it doesn't use strength, I don't think you can replace it with dexterity.

Quote from DxJxC >>

I think it is probably still CON, now that you pointed it out.

The damage die definitely increases and you can stunning strike with it and use bonus action martial Arts/flurry of blows after it. But since it doesn't use strength, I don't think you can replace it with dexterity.

 Yep it's still a CON attack.