Let me answer the question that I believe you are trying to ask. Namely, if we can make a regular cardinal $\kappa$ into a singular cardinal $\kappa$ by forcing of size at most $\kappa^+$, without collapsing any cardinals, must $\kappa$ be measurable?
The answer is no.
The reason is that it is consistent (relative to the existence of a measurable cardinal) with ZFC tat there is a non-measurable cardinal $\kappa$ that becomes measurable in a forcing extension, by forcing to add a Cohen subset to $\kappa$. This is explained in my answer to the question Can measures be added by forcing? Furthermore, one can arrange in that argument that th GCH holds, and that there are no other measurable cardinals.
So suppose that $V$ satisfies ZFC+GCH and there are no measurable cardinals in $V$, but $\kappa$ becomes measurable in $V[g]$, where $g$ was $V$-generic for the forcing to add a Cohen set $g\subset\kappa$. This does not collapse cardinals. Since $\kappa$ is measurable in $V[g]$, we may now perform Prikry forcing over $V[g]$ to add a Prikry sequence $s$, which changes the cofinality of $\kappa$ to $\omega$, but does not collapse cardinals.
So in $V$, there were no measurable cardinals and $\kappa$ was regular, but the combined forcing $g\ast s$ made $\kappa$ into a singular cardinal without collapsing any cardinals. This combined forcing has size $\kappa^+$ under the GCH.