Without loss of generality, let the distance between $A$ and $B$ equal 3 (this is for convenience in calculations) and suppose that $\overline{AB}$ is horizontal (this just makes it easier to talk about where things are).

Now, the point $P_1$ that is on $\overline{AB}$ a distance of 2 from $A$ and 1 from $B$ is in the locus, as are points $P_2$ and $P_3$ that are directly above and below $B$, a distance of $\sqrt{3}$ from $B$ and $2\sqrt{3}$ from $A$. Using these three points, we can determine that if the locus were a circle, its center would have to be at a point we'll call $C$, on $\overrightarrow{AB}$ a distance of 4 from $A$ and 1 from $B$ (1 unit past $B$ from $A$).

Next, consider a point $P$ in the locus with $PA=2x$, $PB=x$, and $PC=y$. Apply Stewart's Theorem to get $\begin{align} 3y^2+(2x)^2&=4(x^2+3) \\ 3y^2+4x^2&=4x^2+12 \\ y^2=4 \\ y=2. \end{align}$ So, all possible points $P$ in the locus are a distance of 2 from $C$, which means that the locus is a circle of radius 2, centered at $C$.
n.b. This suggests that if $A=(a,0)$ and $B=(-a,0)$, $C=(-2a,0)$ and the radius is $2a$, which would give an equation of $(x+2a)^2+y^2=4a^2$, which is different from what you said you got.