Expectation of normal RV conditional on normal mixture

Let $$v\sim\text{Normal}\left(\mu,\sigma_v^2\right)$$ a random variable with $$\mu>0$$ and $$u\sim\text{Normal}\left(0,\sigma_u^2\right)$$

Let $$k\sim\text{Binomial}\left(N,p\right)$$ a random variable with $$N\geq 2$$ an integer and $$p\in\left[0,1\right]$$.

Finally, let $$q= k\times\beta\times\left(v-\mu\right)+u$$.

What is $$\mathbb{E}\left(v\mid q\right)$$?

I computed $$\mathbb{E} q = 0$$, $$\text{var} \left(q\right) = \sigma^2_u + \beta^2 \sigma^2_v \times \left[p \left(N(1-p) + N^2 p\right)\right]$$, and $$\text{covar}\left(q,v\right)=\beta \sigma^2_v N p$$ (and verified the results numerically).

I don't think one can use in this case the following regression (??): $$\mathbb{E}\left(v\mid q\right) = \mu+\frac{\text{covar}\left(q,v\right)}{\text{var} \left(q\right)} q,$$ since $$q$$ is essentially a mixture of normals ($$k$$ is random).