Straight up, White Haired Witch with Strength Patron is how this combo comes to be. Divine Power is the level 8 spell.
There are natural attack cleric options though too, so how does it all break down? Do you get the extra primary attack? It would lose its 1.5× even if it was your only because you're making multiple attacks. Tnat seems to me what makes sense, but I could be wrong.
You get an extra attack.
The 1.5x STR is based on how many natural attacks you have (i.e. only having a bite) not on how many attacks you make in a round. Under most circumstances, it is the same (if you only have a bite, a bite is all you can take, even with a full attacks action) but that isn't true with a haste (or similar) attack.
So you could take 2 natural attacks with you single primary natural attack and get 1.5 str on both.
Huh... I guess I got confused by "If a creature has only one type of attack, but has multiple attacks per round, that attack is treated as a primary attack, regardless of its type." I guess that doesn't really conflict with the 1.5× statement though. Dunno why I thought it did.
Makes mono-natural attack Shifters like the adaptive or bull aspect quite a bit more appealing though.