Definition. A monad over a projective variety $X$ is a complex $$M : 0 \longrightarrow \mathcal{A} \stackrel{f} {\longrightarrow} \mathcal{B} \stackrel{g} {\longrightarrow} \mathcal{C} \longrightarrow 0$$ of coherent sheaves over $X$ which is exact at $\mathcal{A}$ and at $\mathcal{C}$, that means $g \circ f = 0$, $f$ is injective and $g$ is surjective. The coherent sheaf $E : = \dfrac{\text{Ker}(g)}{Im(f)}$ will be called cohomology of $M$.
Consider the monad $M$ and the exact sequences associated with it.
$$M : 0 \longrightarrow \mathcal{O}_{\mathbb{P}^{3}}(-1) \stackrel{f} {\longrightarrow} \mathcal{O}_{\mathbb{P}^{3}}^{\oplus 4} \stackrel{g} {\longrightarrow} \mathcal{O}_{\mathbb{P}^{3}}(1) \longrightarrow 0$$
$$0 \longrightarrow \mathcal{O}_{\mathbb{P}^{3}}(-1) \longrightarrow K \longrightarrow E \longrightarrow 0 \tag{1}$$
$$ 0 \longrightarrow K \longrightarrow \mathcal{O}_{\mathbb{P}^{3}}^{\oplus 4} \longrightarrow \mathcal{O}_{\mathbb{P}^{3}}(1) \longrightarrow \tag{2}0 $$
$$0 \longrightarrow \mathcal{O}_{\mathbb{P}^{3}}(-1) \longrightarrow \mathcal{O}_{\mathbb{P}^{3}}^{\oplus 4} \longrightarrow Q \longrightarrow 0 \tag{3}$$ and
$$0 \longrightarrow E \longrightarrow Q \longrightarrow \mathcal{O}_{\mathbb{P}^{3}}(1) \longrightarrow 0 \tag{4}$$
where $K = \text{Ker}(g)$, $Q = \text{coker}(f)$ and $E$ is the cohomology of $M$.
In this case, we have that $E$ is a instanton sheaf of charge $c = 1$. As your dual $E^\vee$ is a reflexive sheaf and $E \simeq E^{\vee}$ we have that $E$ is a reflexive sheaf.
The goal here is to calculate $\text{dim}Hom \bigl( E \otimes T_{\mathbb{P}^{3}} \bigr) = h^{0}(E^{\vee} \otimes T_{\mathbb{P}^{3}}) = h^{0}(E \otimes T_{\mathbb{P}^{3}})$.
My attempt.
By [I, see proposition 19] we have $E$ is locally free and stable.
Twisting the Euler sequence by $E$, we get $$ 0 \longrightarrow E \longrightarrow E(1)^{\oplus 4} \longrightarrow E \otimes T_{\mathbb{P}^{3}} \longrightarrow 0 \tag{5}$$
By [II, see Lemma 1.2.5] we have $H^{0}(\mathbb{P}^{3}, E_{norm} = E) = 0$, because $c_{1}(E) = 0$.
Now, by [III, see Corollary 3.3] we have that $E$ is $1$-regular. So $H^{i}( E(1-i)) = 0$ for all $i > 0$.
Even with this information and using the exact sequences (1), (2), (3) and (4), I still haven't been able to reach the goal mentioned above.
Any help is most welcome.
Thank you very much.
I) Instanton Sheaves on Complex Projective Space. (Marcos Jardim),
II) Vector Bundle on Comlex Projective Spaces. (Okonek),
III) Monads and Regularity of Vector Bundles on Projective Varieties. (M.Miró-Roig).
Since $E$ is 1-regular, it is $m$-regular for all $m>0$. Then, $h^0(E)=\chi(E)$ and same for $E(1)$. From your (5), since $H^1(E)=0$, we get, $h^0(E\otimes T_{\mathbb{P}^3})=4 h^0(E(1))-h^0(E)=4\chi(E(1))-\chi(E)$. Using (1) and (2), you can easily calculate these.