Most constructs in Monster Core are immune to spirit damage, but there are a couple of exceptions. The soulbound doll says "Because of its soul sliver, a soulbound doll is not immune to spirit as most constructs are", and the sidebar on soulbound homunculi says that they lose the immunity to spirit. So this precedent seems to be that soulbound constructs are not immune to spirit damage, which makes sense.
However, a quick look through Demiplane Nexus has shown that these are not the only constructs that interact with souls. The Radiant Warden from Bestiary 2, ancient harrowkin from Stolen Fate, the Grand Defender from Lost Omens: Highhelm, the Soulbound Ruin from Age of Ashes, and some Divine Wardens mentioned in their Bestiary 3 sidebar are all mentioned to contain souls in some form or another (with the Grand Defender also having the celestial trait). In addition, the Azmakian Effigy from Blood Lords has both the construct and fiend traits (although, unlike most fiends, it is not weak to good) and is mentioned in its sidebar to have a demon bound in it, and the Levaloch from Bestiary 3 has the construct, devil, and fiend traits and says "Levalochs are strange amalgamations of devil and automaton, never requiring food or rest."
If I were to convert these creatures to Remaster rules, should I not give them the usual immunity to spirit damage? I'm thinking most of these creatures wouldn't get the immu itu, but I'm not sure about the effigy and the levaloch.