I have a party of 4, level 13 players. One is a Paladin who has an aura that prevents being frightened as well as a +1 Sword. Two characters can do fire damage. The rest of the party is a Druid, Sorcerer, and Bard. They wipe out solo monsters quickly; and use Banishment. The Paladin did about 80 points of damage in one attack on a nat. 20, and two of the party members can cast fire-based attacks. I think the Mummy Lord wouldn't last more than 2 rounds solo.
I'm thinking of giving the Mummy gold-plate armor that is fused to the mummy's body that grants fire resistance (removing the fire vulnerability), but adds vulnerability to lightning (gold is conductive) to counter meta-gaming knowledge. That might let it last a couple more rounds.
Adding more monsters to this fight seems best, but not sure how many to add or what type Thematically I was considering regular mummies, an Air Elemental or a couple undead constrictor snakes (looking like Bone Naga but without the lore). Would 2-4 more mummies do it, or is that overkill?
Dude, firstly you're probably doing CR wrong, in that you're using it as a guideline for overall encounter difficulty (when it's not).
For example, a CR15 Mummy Lord is only a
Medium encounter for 4 x 13th level PCs, even within the scope of a standard adventuring day (it only chews up 13,000 of roughly 54,000 XP for 4 x 13ths between a single long rest).
Meaning you could literally throw 4 separate Mummy Lord encounters at this party in a single day (to fill in the encounter budget for the adventuring day), and they should be just fine.
A
Medium encounter is this:
Medium. A medium encounter usually has one or two scary moments for the players, but the characters should emerge victorious with no casualties. One or more of them might need to use healing resources.
At that XP budget, the PCs are expected to win, with no casualties, and only 1 or 2 of them MIGHT need to use healing resources after the battle.
Adding to this is the fact your PCs have magic items (and those items are not included in the base games math) so you should expect them to steamroll this encounter.
Personally I would do the following:
Give the Mummy Lord the following (cursed, because Mummy) magic items:
Bracers of Armor, Ring of Fire resistance, Ring of protection. This gives it AC 20, and resistance to Fire (cancelling out its vulnerability). Curse those items with something nasty to anyone foolish enough who attempts to attune to them that isnt the Mummy.
Give the Mummy Lord
Spirit Guardians on its spell list. Cast it using a 5th level slot ASAP (
before the battle if possible - remember the Mummy has
Divination so it knows the PCs are coming). The Mummys other 5th level slot should be used on
Spiritual weapon (bonus action) plus an action of your choice on round 1, and a 6th level
Harm the following round.
Add 7HD to the Mummy Lord, bringing its total HP up to around 150. Increase its Wisdom to 20 and add +1 to every single Save DC it has (spellcasting and special abilities, plus wisdom based skills and saves).
Those changes increase its CR by 1 to CR 16 and 15,000 XP.
Now add 4
Mummy mooks (AC 15 from their bronzed armor). For flavor, change the weapons to Bronze greatswords (damage stays the same, but is now slashing and necrotic). This adds 2,800 XP to the encounter, pushing it into Hard territory, but shouldn't multiply the difficulty due to the low CR (CR 3) of the Mummies any more than roughly 1.5 (treat the mooks as a single monster).
The total encounter budget is now around roughly 25,000 pushing it into Deadly for that party (and now using roughly 1/2 of that days encounter budget).