Does there exist a totally-ordered-without-endpoints proper class $L$ such that every closed interval in $L$ does have the order type of a closed interval in the Conway's surreal numbers, but $L$ as a whole does not have the order type of Conway's surreal numbers?

In case it helps, the thinking that lead up to my posting this question:

I confess proper classes make me a bit uneasy. So for private intuition I use as a crutch set theory below a strongly inaccessible cardinal $\kappa$. If one considers, as a toy surreal numbers, just Conway's surreal numbers with birthdays less than $\kappa$, then it seems to me that one can then imitate the usual long line construction based on $\kappa^+$.

That said, my understanding says one should find the surreal numbers and in particular the ordinals (as individuals) in ZFC but not the totality of surreal numbers. Perhaps that means one can't define any proper class that will play the role of $\kappa^+$ and fulfill my intuition.