I am not entirely sure you have correctly stated the logical ordering of what you want to ask.

Your question appears to be: suppose $A$ is a finite-dimensional unital CBA, and suppose it has an approximate diagonal satisfying certain conditions; does the unique diagonal element $m\in A \otimes A$ satisfy a certain norm bound?

If this is your question then the answer is negative. The reason, as Pietro Majer pointed out, is that if you assume $(m_\alpha)$ satisfies these conditions then the assumption $\Vert m_\alpha -m\Vert\to 0$ implies that $m$ itself satisfies these conditions. So your question becomes

Let $m$ be the diagonal element for $A$ (which in particular forces $a\Delta(m)=a$ and $a\cdot m = m\cdot a$ for all $a\in A$). Assume there is $K>0$ such that $\|a\Delta(m)\|\leq K\|a\|$. Do we have $\Vert m\Vert\leq K$?

This is false because we always have $\Vert a\Delta(m)\Vert=\Vert a\Vert $ but we might have $\Vert m \Vert > 1$. Some explicit calculations of amenability constants can be found in a paper of Ghandehari, Hatamai and Spronk.

I am not entirely sure you have correctly stated the logical ordering of what you want to ask.

Your question appears to be: suppose $A$ is a finite-dimensional unital CBA, and suppose it has an approximate diagonal satisfying certain conditions; does the unique diagonal element $m\in A \otimes A$ satisfy a certain norm bound?

If this is your question then the answer is negative. The reason, as Pietro Majer pointed out, is that if you assume $(m_\alpha)$ satisfies these conditions then the assumption $\Vert m_\alpha -m\Vert\to 0$ implies that $m$ itself satisfies these conditions. So your question becomes

Let $m$ be the diagonal element for $A$ (which in particular forces $a\Delta(m)=a$ and $a\cdot m = m\cdot a$ for all $a\in A$). Assume there is $K>0$ such that $\|a\Delta(m)\|\leq K\|a\|$. Do we have $\Vert m\Vert\leq K$?

This is false because we always have $\Vert a\Delta(m)\Vert=\Vert a\Vert $ but we might have $\Vert m \Vert > 1$. Some explicit calculations of amenability constants can be found in a paper of Ghandehari, Hatami and Spronk: there is a preprint version on the arXiv 0705.4279v2 but be warned that the theorem numbers are different in the final published version.