You do get complementation in the case you mention. Two key facts that you did not state explicitly but which follow easily from your hypotheses are that every unit vector $e_n$ is in the linear span of at most $K$ of the $a_i$ and $b_i$, and the span of $K$ of the $a_i$ and $b_i$ is contained in the span of at most $N=2K^2$ unit vectors. Using these facts, it is not hard to verify that the orthogonal projection $P$ onto $A$ is bounded in the $\ell_1$, which by interpolation and duality gives you what you want. To check boundedness in $\ell_1$, you just have to give a bound on $\|Pe_n\|_1$ that is independent of $n$. But on the span of $N$ unit vectors, the $\ell_1$ norm is dominate by $N^{1/2}$ times the $\ell_2$ norm.
You do get complementation in the case you mention. Two key facts that you did not state explicitly but which follow easily from your hypotheses are that every unit vector $e_n$ is in the linear span of at most $K$ of the $a_i$ and $b_i$, and the span of $K$ of the $a_i$ and $b_i$ is contained in the span of at most $N=2K^2$ unit vectors. Using these facts, it is not hard to verify that the orthogonal projection $P$ onto $A$ is bounded in the $\ell_1$, which by interpolation and duality gives you what you want. To check boundedness in $\ell_1$, you just have to give a bound on $\|Pe_n\|_1$ that is independent of $n$. But on the span of $N$ unit vectors, the $\ell_1$ norm is dominate by $N^{1/2}$ times the $\ell_2$ norm.