It is consistent that the answer is no:
Let $V=L[U],$ where $U$ is a normal measure on a measurable cardinal $\kappa.$ First note that we can apply Prikry forcing over $V$ to change the cofinality of $\kappa$ to $\omega.$
Now we show that there is no forcing of size $\kappa$ changing the cofinality of $\kappa.$ Suppose not. Let $P$ be such a forcing notion and let $G$ be $P-$generic over $V$. By Dodd-Jensen covering theorem for $L[U],$ there exists an $\omega-$sequence $C\in V[G]$ cofinal in $\kappa$ which is a Prikry sequence
for the classical Prikry forcing $P_U$ in $V$ and $V[G]$ is covered by $V[C]$. Then $V[C]\subset V[G],$ and we have $P_U$ is a projection of $P$, so $P$ has size $>\kappa.$
It is also consistent that the answer is yes:
Start with $GCH$ and a supercompact cardinal $\kappa.$ Force with a reverse Eason iteration $(P_{\alpha}, \dot{Q}_{\alpha} : \alpha <\kappa)$ below $\kappa$ where at each step $\alpha<\kappa,$ the forcing is trivial, unless $\alpha$ is inaccessible, in which case we force with $Col(\alpha, \alpha^{++}).$ Finally force with $Col(\kappa, \kappa^{++}).$ By standard forcing arguments a $\lambda-$supercompact embedding $j: V \rightarrow M,$ for some $\lambda>>\kappa^{++}$ extends to some elementary embedding $j^*:V[G][g] \rightarrow M[j(G)][j(g)]$.
Let $U$ and $U^*$ be normal measure on $\kappa$ derived from $j$ and $j^*$ respectively and let $P_U \in V$ and $P_{U^*}\in V[G][g]$ be the corresponding Prikry forcings. Note that $U=U^*\cap V$ and $P_U=P_{U^*} \cap V.$ Let $H^*$ be $P_{U^*}-$generic over $V[G][g]$ and let $H=H^*\cap V.$ It is easily seen that $H$ is $P_U$-generic over $V$. Now note that $V[G*g\times H]\subset V[G*g*H^*],$$V[(G*g)\times H]\subset V[G*g*H^*],$ so $\kappa$ remains a cardinal in $V[G*g\times H]$$V[(G*g)\times H]$. Let $V^*=V[G*g].$ Then $P_U\in V^*$ has size $\kappa$ (since $(\kappa^{++})^V$ is collapset to $\kappa$ in $V^*$) and it changes the cofinality of $\kappa$ to $\omega$ without collapsing $\kappa$ (or any smaller cardinals).
