Suppose that $A$ is a finite dimensional unital commutative Banach algebra and $A\hat{\otimes} A$ is its Projective tensor product by itself. Define $\Delta:A\hat{\otimes} A\to A$ with 
$$\Delta(\sum\_{i=1}^\infty a\_i\otimes b\_i)=\sum\_{i=1}^\infty a\_i b\_i.$$
There is $m\in A\hat{\otimes} A$ with 
$$a\Delta(m)=a,\qquad (a\in A)$$
Also there is a net $\{m\_\alpha\}\subset A\hat{\otimes} A$ such that $\|m\_\alpha-m\|\to 0$ and there is $K>0$ such that 
$$a\Delta(m\_\alpha)\to a,\qquad \|a\Delta(m\_\alpha)\|\leq K\|a\|,\qquad \|a\cdot m\_\alpha-m\_\alpha\cdot a\|\leq K\|a\|,\qquad (a\in A)$$
Could we conclude that $K-\|m\|\geq C$, for some non-negative real number $C$ ?



Special bound of net and inequality
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](https://arxiv.org/abs/0705.4279v2) but be warned that the theorem numbers are different in the final published version.


