Figuring out the level caps is a simple (but time-consuming) process. It used to be a lot more tedious before /checkparam became a thing.
First step is to remove all Beast Affinity merits, then call the pet with an iLvl 119 Axe equipped to figure out what its natural cap stats are (remove all other gear and take into consideration any Pet:Stat+ from Axe/Job Points).
Second step is to merit Beast Affinity 5/5 and then use Ankusa Gloves/+1 in combination with a variety of iLvl Axes to determine what the pet's stats look like at a variety of levels. We have 11 different iLvls that we can check directly (99, 100, 101, 105, 106, 109, 113, 115, 117, 118, 119), though I don't have Budliqa +1 so I can't directly test iLvl 118. If the jug pet's natural cap stats match one of these levels, you're done.
If you haven't found a match by this point, you at least have narrowed it down to a particular range and there's an additional step that can be used. In the recent case of Warlike Patrick (who caps at iLvl 104) there currently is no iLvl Axe we can use to summon an exact match to the natural cap version, but we can use Max HP to work it out. I always check the Max HP at level 99/natural cap/119 by using Healing Salve in Abyssea (lower pet's health to < 50% then use Healing Salve and multiply the amount healed x 2). Something to keep in mind for this, though there are a couple of exceptions, but for the majority of pets there is no increase in Max HP at level 100. So, for Warlike Patrick:
Level.........Max HP
99...........5372
100...........5372
Natural Cap.....5400
119............5508
The average Max HP increase from level 100 -> 119 is +7.1 HP per Level.
We know from step 2 that the natural cap stats are greater than level 101 and less than level 105, So when we fill in the blanks, we see...
Level............Max HP
99................5372
100...............5372
101...............5379.1
102...............5386.2
103...............5393.3
104...............5400.4 <- There's the natural cap!
...
119................5508
So yeah, the process is more or less like that.
The cap for some pets like BrainyWaluis and GenerousArthur are quickly identified, and others like Warlike Patrick need additional investigation, but it's relatively simple. Maybe someday they'll follow through and actually include the iLvl info in the jug description!