One can omit $0^\sharp$, as well as the need for any large cardinals, if one simply limits the class of forcing extensions. For example, perhaps one wants to consider only amalgamation and non-amalgamation in the generic multiverse of Cohen-real extensions $L[c]$, or in the generic multiverse of proper-forcing extensions $L[G]$ or $\omega_1$-preserving forcing extensions. 

**Theorem.** If $\omega_1^L$ is countable in $V$, then there are $L$-generic Cohen reals $c$ and $d$, whose corresponding extensions $L[c]$ and $L[d]$ are not amalgamated by any $L$-generic Cohen real $e$. Indeed, there are such $c$ and $d$ for which there is no $\omega_1$-preserving extension $L[G]$ extending both $L[c]$ and $L[d]$.

**Proof.** Assume $\omega_1^L$ is countable in $V$. In particular, in $V$ we can enumerate all the dense subsets of Cohen forcing $\text{Add}(\omega,1)$ in $L$ in an $\omega$-sequence in $V$. From this enumeration, we can easily build $L$-generic Cohen reals. Using the proof of proposition 3 in the question, we can code any real coding $\omega_1^L$, and this real cannot be added in any $\omega_1$-preserving forcing over $L$. $\Box$.