= Solution
Work on $\mathbb R$ with <Lebesgue measure>. Without a uniform $L^p$ bound, use
$$
\boxed{f_n=n^{-1}1_{[0,n^q]}.}
$$
It has $\|f_n\|_q=1$ and $\|f_n\|_r=n^{q/r-1}\leq1$, with $1/r=0$ for $r=\infty$. Each function still belongs to $L^p\cap L^r$, but its $L^p$ norm is unbounded. For every fixed $\varepsilon>0$, its level set above $\varepsilon$ is eventually empty, defeating any positive uniform $M$.
Without a uniform $L^r$ bound, use instead
$$
\boxed{g_n=n\,1_{[0,n^{-q}]}.}
$$
Now $\|g_n\|_q=1$ and $\|g_n\|_p=n^{1-q/p}\leq1$, but for any fixed $\varepsilon>0$ the high-amplitude level set has measure $n^{-q}\to0$. These give the two distinct failures of the <nonvanishing level-set bound between three Lp norms> when either control is removed.
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