Your algebra is Type I and residually finite dimensional (RFD) Therefore it's unitization is also Type I and RFD. Let's call $B$ it's unitization. Since $B$ is Type I it satisfies the UCT and therefore embeds into a unital simple AF algebra by Huaxin Lin's 2000 paper in Proc. AMS (RFD algebras and AF embeddings).
You may also want to check out Marius Dadarlat's 2000 paper "Nonnuclear subalgebras of AF algebras." His restricted Bratelli diagrams give a very nice straightforward way to embed any unital RFD nuclear C-algebra into a unital simple nuclear monotracial C-algebra.