If a level 7 Beast Master Ranger were to give their Beast of the Land Primal Companion barding armor, what is the new Armor Class? For example let's use chainmail barding. It seems that the new AC would be calculated as 16 + 3 (ranger's PB). Is this right?
It no longer stacks, but it used to
There is a subtle but substantial difference between the old Ranger's Companion and the newer Primal Companion.
Let's take a look:
Ranger's Companion: [...] Add your proficiency bonus to the beast’s AC, attack rolls, and damage rolls, as well as to any saving throws and skills it is proficient in. [...]
As you can see, it specifically says you add the PB to the creature's AC.
Let's take the Black Bear as an example. By its stat block, it's AC is 11 (Natural Armor). A Black Bear with Chainmail barding would have AC 16. Since the feature says to add your PB to the beast's AC you would get a final AC of 16+PB for a black bear wearing chainmail, against an AC of 11+PB for a "naked" black bear.
Now let's look at Primal Companion:
Primal Companion: [...] Choose its stat block-Beast of the Land, Beast of the Sea, or Beast of the Sky-which uses your proficiency bonus (PB) in several places. [...]
Beast of the Land
Armor Class: 13 + PB (natural armor)
As you c an see, there is a huge difference. The PB here is not added as a bonus to AC, but rather is part of the formula to calculate the creature's AC.
What barding (and armor in general) does is change the formula (specifically, Chainmail gives you a flat amount of 16), which means the PB simply does not come into play anymore.
As Thomas already pointed out, armor class rules specify that:
If you have multiple features that give you different ways to calculate your AC, you choose which one to use.
Which in your case means you can choose which formula to use between "AC: 13 + PB (natural armor)" given by the beast of the land or "AC: 16" given by chainmail.
Choose either 16 or 13+PB
The Primal Companion’s Natural Armor feature sets the companion’s armor class to 13+PB. However, chain mail sets your armor class to 16, and does not interact with other modifiers:
|Chain Mail||75 gp||16||13||Disadvantage||55 lbs|
As you can see, the AC for chain mail is 16 with no other modifiers. It sets your AC to a flat 16. The rules for armor class state:
Some spells and class features give you a different way to calculate your AC. If you have multiple features that give you different ways to calculate your AC, you choose which one to use.
Chain mail provides the companion with a different way to calculate its AC, so you must choose which way to use, either 13+PB or 16. Unfortunately, there is no way to add your PB to the 16 AC from chain mail. It sets your companion’s AC to a fixed value. Only the Natural Armor feature includes the PB as a bonus to AC.