Here is a counterexample of size 38.

Let $m=8$ and let $$P=\{0,a_1,\dots,a_m,1\}\cup\{b_{ij}: 1\le i<j\le m\}$$
where $0<a_i<b_{jk}<1$ whenever $i$ is distinct from $j$ and $k$.

The cardinality of $P$ is $n=1+m+\binom{m}{2}+1=38$.

The join-irreducible elements are only the $a_i$ and $0$, since
$b_{ij}=\bigvee\{a_k:k\ne i, k\ne j\}$.

The $a_i$ each have $\binom{m}{2}-m=20>19=n/2$ elements above them.