No. What follows appears to be a counterexample for $C = \text{Vect}$ (I don't understand where in your argument you prove fullness).

Let $M = \text{Vect}^{op}, C = \text{Vect}$, and let $h : \text{Vect}^{op} \to \text{Vect}$ be the contravariant functor $V \mapsto V^{\ast} \cong \text{Hom}(V, k)$. If smallness is important to you pretend that the first $\text{Vect}$ means vector spaces of at most countable dimension. The endomorphism ring of $h$ is $k$, so the lift $\tilde{h}$ is just $h$ again. 

I claim that $h$ is not full. To see this, if $V$ is a countable-dimensional vector space, regarded as an object in $\text{Vect}^{op}$, then the induced map

$$\text{End}(V, V) \to \text{End}(V^{\ast}, V^{\ast})$$

has the property that its source has dimension $\aleph_0^{\aleph_0} = 2^{\aleph_0}$, but its target has dimension at least $\left( 2^{\aleph_0} \right)^{2^{\aleph_0}}$.

(Taking the opposites of familiar abelian categories seems to be my <a href="http://mathoverflow.net/questions/143807/is-the-morphism-coproduct-product-in-additive-category-monic/143816">new</a> <a href="http://mathoverflow.net/questions/103252/why-are-injective-modules-more-complicated-than-projective-modules">favorite</a> trick! Note that vector space duality establishes that $\text{FinVect}$ is equivalent to its opposite. $\text{Vect}$ itself is the ind-completion of $\text{FinVect}$, so $\text{Vect}^{op}$ is the pro-completion; in other words, it's the category of provector spaces. This category is also known as the category of linearly compact vector spaces: see <a href="http://mathoverflow.net/questions/104777/what-are-the-algebras-for-the-double-dualization-monad">this MO question</a> and <a href="http://golem.ph.utexas.edu/category/2012/09/where_do_linearly_compact_vect.html">this n-cafe post</a> for details. This gives some intuition for why $h$ is not full: it's the forgetful functor and it doesn't see the topology.

In the special case that $k = \mathbb{F}_p$ we can be a little more explicit: $\text{Vect}^{op}$ in this case is the full subcategory of profinite groups consisting of the ones whose finite quotients are all elementary abelian $p$-groups.)