Let $\:\langle \hspace{.02 in}\mathbf{U},\hspace{-0.03 in}\mathcal{T}_u\rangle\:$ be a Hausdorff space. $\;\;\;$ Let $\hspace{.02 in}\Gamma\hspace{.02 in}$ be a closed subset of $\hspace{.04 in}\mathbf{U}^{\hspace{.02 in}3}$ such that
for all elements $\:x,\hspace{-0.03 in}y\hspace{-0.02 in},\hspace{-0.03 in}z,\hspace{-0.04 in}w\:$ of $\hspace{.02 in}\mathbf{U}$, $\;$ if $\:\langle x,\hspace{-0.03 in}y\hspace{-0.02 in},\hspace{-0.03 in}z\rangle \in \Gamma \:$ and $\:\langle x,\hspace{-0.03 in}y\hspace{-0.02 in},\hspace{-0.03 in}w\rangle \in \Gamma \:$ then $\:z=w$
and
for all open subsets $V$ of $\hspace{.02 in}\mathbf{U}$, $\;\; \left\{\langle x,y\rangle \in \mathbf{U}^{\hspace{.02 in}2} \: : \: (\exists z)\left((z\in \hspace{-0.02 in}V\hspace{.03 in})\:\text{ and }\:\left(\langle x,\hspace{-0.03 in}y\hspace{-0.02 in},\hspace{-0.03 in}z\rangle \in \Gamma\hspace{.03 in}\right) \hspace{.02 in}\right)\right\} \;\;$ is open
and
for all elements $\:x,\hspace{-0.03 in}y\hspace{-0.02 in},\hspace{-0.03 in}z,\hspace{-0.03 in}a,\hspace{-0.03 in}b\hspace{-0.02 in},\hspace{-0.03 in}c\:$ of $\hspace{.02 in}\mathbf{U}$, $\;$ if $\: \langle x,\hspace{-0.03 in}y\hspace{-0.02 in},\hspace{-0.03 in}a\rangle \in \Gamma \:$ and
$\langle a,\hspace{-0.03 in}z\hspace{-0.01 in},\hspace{-0.02 in}b\rangle \in \Gamma \:$ and $\: \langle \hspace{.02 in}y\hspace{-0.02 in},\hspace{-0.03 in}z,\hspace{-0.03 in}c\rangle \in \Gamma \:\:$ then $\:\: \langle x\hspace{-0.01 in},\hspace{-0.03 in}c,\hspace{-0.03 in}b\rangle \in \Gamma$
and
there exists an element $e$ of $\mathbf{U}$ such that for all elements $x$ of $\mathbf{U}$, $\: \langle e\hspace{-0.01 in},\hspace{-0.02 in}x,\hspace{-0.02 in}x\rangle \in \Gamma \:$ and $\: \langle x,\hspace{-0.03 in}e,\hspace{-0.02 in}x\rangle \in \Gamma$
and
for that element $e$ and all elements $x$ of $\mathbf{U}$, there exists an
element $y$ of $\mathbf{U}$ such that $\: \langle x,\hspace{-0.03 in}y\hspace{-0.01 in},\hspace{-0.02 in}e\rangle \in \Gamma \:$ and $\: \langle \hspace{.03 in}y\hspace{-0.02 in},\hspace{-0.02 in}x,\hspace{-0.03 in}e\rangle \in \Gamma$
and
for that element $e$, the map $\; \operatorname{inv} : \mathbf{U} \to \mathbf{U} \;$ given by $\; \langle x,\hspace{-0.02 in}\operatorname{inv}(x),\hspace{-0.02 in}e\rangle \in \Gamma \;$ is continuous
.
(The uniqueness of $e$ follows from the $\:\: z=w \:\:$ condition, and the uniqueness of $y$ such that
$\langle x,\hspace{-0.03 in}y\hspace{-0.01 in},\hspace{-0.02 in}e\rangle \in \Gamma \:$ follows from the condition asserting its existence and the two conditions before that.)
Does it follow that there exists a $\hspace{.03 in}\big(\hspace{-0.01 in}$T$_0$$\hspace{-0.02 in}\big)\hspace{.01 in}$ topological group $\:\langle G,\hspace{-0.03 in}\cdot,\hspace{-0.04 in}\mathcal{T}_g\rangle \:$,
an open subset $V$ of $\hspace{.02 in}\mathbf{U}$,$\hspace{.02 in}$ an open subset $W$ of $G$ such that $G\hspace{.02 in}$'s identity
element is in $W\hspace{-0.04 in}$,$\hspace{.02 in}$ and a homeomorphism $\: \phi : V\to W \:$ such that
for all elements $\:x,\hspace{-0.03 in}y\hspace{-0.02 in},\hspace{-0.03 in}z\:$ of $V\hspace{-0.03 in}$, $\;\;\; \langle x,\hspace{-0.03 in}y\hspace{-0.02 in},\hspace{-0.03 in}z\rangle \in \Gamma \;$ if and only if $\:\: \phi(x) \cdot \phi(\hspace{.03 in}y) \: = \: \phi(z) \;\;$?
I did not go deeply into your definition of a local group because they are not among my mathematical interests, but in [$\S$ 23.H] of the second Russian edition of the classical Lev Pontrjagin’s book “Continuous groups” is written that each local Lie group is locally isomorphic to a topological group (and the proof of this fact uses very complex analytic methods, see $\S$ 59), but there exists a local group which is not locally isomorphic to a topological group (see Lie S. und Engel F., Theorie der Transformationsgruppen. Leipzig, Teubner, 1, 2, 3).
In this answer are links to Pontrjagin’s book.