You do not need 4 tanks. There are two ways you can handle this.

1: Make sure the second tank that gets called pays attention to if the spell "Imminent Destruction" is actually cast on them. Sometimes a second tank gets called but the spell is never actually cast. This means the second tank called can usually ignore it. Look for a boot with lightning bolts striking into it as the icon in the detriment window. I get called often as the other tank is called and 90% of the time, only one tank ever really gets hit with the detriment.

2: If both tanks do actually get the detriment, the 3rd tank can hold both for a short time if heals and debuffs are maintained.

We use 3 tanks; Zerker on Caden, Crusader on Keplin, and a Brawler there to snap one when a tank needs to joust out.

We did our first pull on this, and tried having the DPS follow the named that wasn't on the mount. We seemed to have a number of times where both nameds would call out for a tank swap.

I think this happened due to the mount swapping between the nameds such that:
1. Caden called out for tank swap
2. Mount changed over to Caden
3. Keplin called out for tank swap

Unfortunately, we didn't have four tanks ready to roll to handle this. Is there something we're missing here in terms of mount/dismount mechanic? Should we perhaps stay on the mounted mob until the tank swap is over to buy some time?

Do not split them. In positioning or in your raids DPS. DPS needs to all focus back and forth. Always burn the one that is NOT on the mount. When the mount swaps, switch mobs. Also, if you pull the two named too far apart, they will cast a buff on eachother that pretty much = a fail condition.

What's the best tactic? All raid to the same named or split raid in the two nameds?

This fight is not a DPS check. It is a patience and coordination fight. Obviously the more DPS you have the faster it will be over though. 25 million should be more than enough... altho this fight will make it feel like its about 5 million.