Can we split the difference and have one kid?

You can't have half a kid. This is a core incompatibility, not a negotiable issue. Either you have kids (wanted parent gets what they want; reluctant parent is resentful) or you don't (wanting parent is resentful). Research shows relationships where one partner is ambivalent about parenting result in either divorce or deeply unhappy families.

Is it possible someone will change their mind?

Maybe, but betting the relationship on it is risky. Some ambivalent people become excited parents; many don't. Some childfree people get baby fever; many don't. You can have honest conversations about whether minds might change, but you can't build a relationship on hope. Clarity is essential: "Are you open to possibly wanting kids?" not "I'll wait until you change your mind."

Should we stay together if we disagree?

Only if one person genuinely becomes certain they don't want kids (want becomes don't) or becomes open to wanting them (don't becomes maybe). Otherwise, staying is unfair to both: one never gets kids; one gets kids from unwilling partner. It's one of the few true dealbreakers in compatibility.