Singleton in a simple SBM
This answer shows an approach to the problem in the question.

(Threshold of isolated vertices) Let $G(n,p)$ be a random graph with $n$ vertices and each edge is present with probability $p$ independently. Suppose that $p\le c{\log n}/n$ for some constant $c$. Then the possibility of the appearance of at least one isolated vertex in $G(n, p)$ goes to 1 when $n$ goes to infinity.  

Assume the above proposition on the threshold of isolated vertices holds. Applying it to the case of $G(2n, p+q)$ where $p+q\le c \log (2n)/2n$, we know the appearance of at least one isolated vertex goes to 1 when $2n$ goes to infinity.

Since the random graph in the question has less edges than the random graph above, the appearance of at least one isolated vertex in the random graph in the question also goes to 1 when $n$ goes to infinity. (This loose argument can be made rigorous.)

If you prefer not to use that proposition, you can transplant its proof to fit the current problem. It should not be hard to find a proof for that proposition since it is a classic result in the introductory theory of random graphs.