The
Mr Roboto topic has a short discussion, and links to
another discussion, on using magic jar on a construct. Rather than reply to a 2+ year old thread, I'm making this one.
I know it's been a long time, but there's no immunity to the magic jar effect, at least not from immunity to mind affecting. The takeover attempt is not mind affecting, nor is magic jar itself. (Interestingly, you may not activate (ex) or (su) abilities, and spells and SLA's stay with the hosts own soul.) However, constructs are immune to necromancy effects, which magic jar definitely is. From the linked discussion, I'd say that lowering immunity is indeed possible, given that elves can choose to be affected by a sleep effect.
Mind switch would not work on its own, since it's mind affecting, and mind affecting effects only work against creatures with an intelligence score of at least 1.
This is not immunity or resistance, but a failure of the effect to function due to illegal targeting. Now, following the psionic sandwich formula, you'd be able to PAO the effigy into a creature with an Int score (unlike polymorph, PAO alters mental stats), use [true] mind switch, then wait for the PAO to expire (or dispel it). PAO is legal, due to its ability to affect objects.
In conclusion, the magic jar works for the trick, mind switch is indeed better, but takes some finagling.