2 Drastic cut, and postponement of answer.

Now,

I think started to write a complete answer here and accidentally hit the "first" example in which there is multiplicity greater than $1$ is the $A_1 \times A_1$ Levi in the simply connected simple split group of type $A_3$. Let $K$ be our $p$-adic field and $R$ its valuation ringPost" button. Let $G$ be the connected reductive group over $K$ which fits into the following exact sequence:$$1 \rightarrow G \rightarrow GL_2 \times GL_2 \rightarrow G_m \rightarrow 1,$$where the penultimate arrow sends $(g_1, g_2) \in GL_2 \times GL_2$ It seems more difficult than I thought to $det(g_1) \cdot det(g_2)$cook up an example. Let $Z$ denote the center of $GL_2$, so $Z$ is isomorphic I'll try to $GL_1$.

Then $G_\circ$ consists of pairs $(g_1 z_1, g_2 z_2)$ such that $g_1, g_2 \in SL_2(K)$think more about it, $z_1, z_2 \in Z(R)$ and $\det(g_1 z_1) \det(g_2 z_2) = 1$.

Let $\pi$ be I'll post at a supercuspidal irrep of $G$. Let $\tilde \pi$ denote the induction of $\pi$ to $GL_2 \times GL_2$. Let $\Delta: GL_2 \rightarrow GL_2 \times GL_2$ denote the diagonal embeddinglater time/date.

1

First, for generalities related to the question, I'd recommend the article of Bushnell and Henniart, " Generalized Whittaker Models and the Bernstein Center," in Amer. J. of Math., Vol. 125, No. 3, Jun. 2003, pp.513-547. Starting in Section 8, you can find a nice and explicit survey and expansion on the results of Bernstein. Some of the "what do I know" facts you mention can be found in this paper of Bushnell-Henniart (see Example 8.6).

Now, I think the "first" example in which there is multiplicity greater than $1$ is the $A_1 \times A_1$ Levi in the simply connected simple split group of type $A_3$. Let $K$ be our $p$-adic field and $R$ its valuation ring. Let $G$ be the connected reductive group over $K$ which fits into the following exact sequence: $$1 \rightarrow G \rightarrow GL_2 \times GL_2 \rightarrow G_m \rightarrow 1,$$ where the penultimate arrow sends $(g_1, g_2) \in GL_2 \times GL_2$ to $det(g_1) \cdot det(g_2)$. Let $Z$ denote the center of $GL_2$, so $Z$ is isomorphic to $GL_1$.

Then $G_\circ$ consists of pairs $(g_1 z_1, g_2 z_2)$ such that $g_1, g_2 \in SL_2(K)$, $z_1, z_2 \in Z(R)$ and $\det(g_1 z_1) \det(g_2 z_2) = 1$.

Let $\pi$ be a supercuspidal irrep of $G$. Let $\tilde \pi$ denote the induction of $\pi$ to $GL_2 \times GL_2$. Let $\Delta: GL_2 \rightarrow GL_2 \times GL_2$ denote the diagonal embedding.