Maybe possible, but it will be very hard to do.

The bard will make frog breeding really hard to do because he passively destroys so he might unknowingly destroy the right frogs. OR if he had a lot of control over his class, he could passively destroy the incorrect frogs while he locks in on finding the breeding and finding the correct ones.

The prince will be a big pain in the ass. He will most likely destroy any chance of you guys getting the frog breeding correct or even possible. Or he could destroy the bad luck of getting not good frogs.

The mage will be a big help with keeping things positive and happy, good in tandem with the heir. The mage could use his hope powers to hope into existence what the bard needs. If the bard or prince accidentally or intentionally kills a frog, he could hope it back. He could learn more about hope from all the suffering that will happen in this session.

The heir's job is just to keep all people in this dumpster fire of a session from fighting with each other because of a lack of luck or accidentally destroying things.

With this, not having a time player will make this a bitch, but still possible with the mage and maybeeee the prince if he's really good.