The Boolean algebra of all almost invariant subsets of an uncountable locally finite group is contained in every Sub-Boolean that separates points Let $G$ be a group. A subset $A\subset G$ is said to be almost right invariant if $A\Delta A\cdot g$ is finite for all $G$. The family of all almost right invariant subsets $\mathcal{B}_G$ of $G$ is a sub-Boolean algebra of the power set  $\mathcal{P}(G)$.
Assume that $G$ is uncountable locally finite group, and $\mathcal{B}$ is a sub-Boolean algebra of the power set  $\mathcal{P}(G)$ that separates points, (that is for all $g\neq h\in G$ there exists $A\in \mathcal{B}$ such that $g\in A$ and $h\in A^c$).
I would like to prove that $\mathcal{B}$  contains a sub-Boolean algebra that is isomorphic to $\mathcal{B}_G$.
 A: Not if $|G|=2^{\aleph_0}$.
It is a theorem of D. Holt [H] that for an uncountable locally finite group $G$, the Boolean of (right) almost invariant subsets is reduced to that $W_G$ consisting of finite and cofinite subsets in $G$. (This theorem was previously obtained by Scott and Sonneborn [SS] for uncountable locally finite abelian groups.)
Given this, the question becomes equivalent to the following one, unrelated to group theory:

Let $X$ be an uncountable set. Is it true that every Boolean subalgebra of the power set of $X$, separating points, contains $|X|$ pairwise disjoint nonzero elements.

And indeed, on the reals let $A$ be the set of subsets that are finite union of left-closed, right-open intervals. This is a Boolean subalgebra; it separates points but has no uncountable subset of pairwise disjoint elements.
[SS] W. Scott, L. Sonneborn. Translations of infinite subsets of a group. Colloq. Math. 10
1963 217–220
[H] D. Holt. Uncountable locally finite groups have one end. Bull. London Math. Soc. 13
(1981), no. 6, 557–560
A: An extended comment:
Regardless of $X$ being uncountable or not, as long  as it is infinite the Stone space of $W_X$ is the one-point compactification of $X$ (as a discrete space). Knowing that the Stone space of a sub-Boolean algebra of the power set separating points is a compactification $Y$ of $X$; and that for a compactification $Y$ of $X$, the family $\mathcal{B}_Y:=\{A\cap X:A$ is clopen subset of $Y\}$ (assuming that $X\subset Y$) is a sub-Boolean algebra separating  points; and that a compactification $Z$ of $X$ dominates $Y$ if and only if $\mathcal{B}_Z$ contains a sub-Boolean algebra isomorphic to $\mathcal{B}_Y$, we can deduce that $W_X$ is isomorphic to sub-Boolean algebra of any sub-Boolean algebra separating points.
My question is: Is there a direct proof to the fact that every sub-Boolean algebra of the power set of $G$ ($G$ is an uncountable locally finite group) separating points contains a copy of $\mathcal{B}_G$ (the sub-Boolean algebra of all almost right invariant subsets)?
