No answer, just some data. Up to $n=825$ there are 41 pairs $[p,n]$ such that $S_n \equiv 0 \mod p^2$ and $\frac n2 \le lt p \le lt \frac{2n}{3}$. Here they are: $ \small [7, 12], [11, 21], [29, 55], [41, 68], [43, 72], [47, 80], [61, 100], [73, 136], [\mathbf{89}, 138], [\mathbf{89}, 150], [79, 156], [89, 167]$ $ \small [109, 183] [\mathbf{127}, 206], [\mathbf{127}, 230], [131, 231], [157, 276], [181, 301], [199, 306], [197, 364], [227, 386], [257, 445] $ $\small [\mathbf{277}, 450], [\mathbf{277}, 475] [313, 482], [251, 492], [353, 538], [307, 542], [421, 654], [439, 670], [367, 701], [431, 702]$ $\small [\mathbf{359}, 703], [\mathbf{359}, 710], [373, 731] [401, 737], [467, 737], [409, 755], [431, 757], [491, 798], [419, 822] $
Over the same range the (naively ) expected number of repeat divisors of that type is about $43$ so the result seems almost certainly true, but perhaps for no special reason.
It is notable that several times one gets the same $p$ twice in a row. I don't know if it is significant however.
