Its very unlikely its Slime out of all the DQ characters. Square as we know is extremely stingy with their things, why would they push for a character that is not even the protagonist? Instead its a random character like a Moogle, I get that Slime is popular. However so are Moogles and yet we got Cloud and not a Moogle. I don't see Square negotiating with Nintendo and giving them Slime, it just would not make sense on their part. And it does not fit Square's past business decisions, they do not push for guest characters that are not the protagonist. Its just not how Square does things, theres a reason we have Noctis in Tekken and not a Cactuar.
That isn't why Cloud was chosen. Being a protagonist was meaningless. Being effectively the mascot and most memorable of the franchise is why people wanted him in. Besides, none of the monsters in Final Fantasy have the same mascot status as Slime does. Chocobo and Moogle are important, but Cloud has always trumped them in mascot status. Slime is however vastly more popular than Erdrick, but that's what happens when you're way more known in comparison. Consider the fact that many only know of the first two game outside of Japan. Erdrick doesn't exist outside of lore. Meanwhile, pretty much all know Slime, the actual mascot, who is pushed more than any one Hero(and many of the Heroes aren't notable at all. They're just the player avatar. The kicker is that Slime has his own games. It's also important in the Dragon Quest Monster games, literally being the Pikachu of it. Though unlike Pikachu, it doesn't have this "can't evolve/mate/etc. stuff" going on. Far less situational of a design. But it's not like Slime was based off of a specific version, it's just the first time it had characterization.
Slime is also a protagonist of some games. He's the Pikachu of the series. It makes sense to promote the actual mascot of the series first just as much as a player character. Slime is also more iconic worldwide, making him a very memorable choice. And Square does not only push protagonists as options. Mascots matter too. The noted Mario Hoops 3-on-3 was made by Square-Enix in Japan first. They chose Slime and other general Final Fantasy characters(but not actual protagonists, just generic classes) to work with. In a crossover.
There is no precedent of "player protagonist only" to go off of. They do whatever makes sense. And let's be honest, they're not exactly known for using the actual player characters of DQ over Slime to begin with. Even other monsters have taken precedence. So in the end, it more comes down to what Nintendo wants more. I could see them still saying yes to Erdrick.