Congrats on pulling a Permit! I have not hunted Swans in ID, but have pulled a few Permits over the years in ND with some success. They are typically very plentiful in the area we hunt ducks/geese and had good success just pass shooting them.
I’ve never hunted them either, but when I’ve seen them in ND they seem to have a fairly consistent daily pattern and it seems like pass shooting could be effective.
I've killed and hunted them in Utah, and seen them while duck hunting in southern Idaho while they have flown right over me. I've called them in by just verbally "whooping" even without decoys. They taste good in the slow cooker, make sure you use the biggest shells you have- I lost a swan because I had 3" in instead of my 3 1/2" BBs.
We have killed a bunch here in Sodak. Put out a couple snow goose floaters and they decoy right in. Two swan floaters would probably work even better. We shoot them with 2 shot over decoys