Yep. If you don't restrain the post cover from expanding the foam could blow the joints open. You'd need to clamp in several directions or use a band clamp that wouldn't stretch at all.
It's not clear from your post whether you have access above the ceiling drywall. If not, how are you injecting the foam and monitoring to make sure you don't put in too much or too little?
I would assume that you've built the cover in two halves, with minimal clearance to the post cover. You could cut an octagonal piece of 1x to fit inside the cover, drill/jigsaw a hole about an 1/8" larger than the post itself in the center, cut the octagonal piece in half and polyurethane glue it to the drywall and post - make sure you have the orientation right. Then just PU glue or caulk the two post cover halves in place.
If the hole wasted the octagonal piece and only triangular tips are left, you could glue those in place the same way. You'd just have to make sure that you attach them in the right places so the column cover closes correctly.
Presumably the eight pieces of wood are mitered so that they mate with each other and they will be adhered together with glue or some such? If so all that should be required is to add some spacer blocks inside the top of each piece of a thickness that will build them out so that the remaining space is just sufficient to surround the round metal post. That will keep the surround from shifting and once that is achieved there should be no need to actually adhere it to the metal post. I'd be just a bit afraid of making the surround too permanent just is case some work or inspection needs to be done on the metal post.
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