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There are a number of sewing machines that can be used to accomplish the tasks you are asking about. To find what you are looking for it is best to know manufacturer and model names.
1) For this item I am reading this as how do you sew the ends of the sweatband together to make it a circular shape. The answer to that is any flatbed sewing machine that can do a zig-zag stitch. However, you still need to do a little hand sewing to keep it together before taking it to the sewing machine, this is especially true if you will be belling the sweatband.
2) For sewing the sweatband into the hat, this is assuming a reeded sweatband you will want to look for a Singer 107-1 or the clone ASM 1107-1. Both are hard to find and from my understanding the ASM is no longer made. You could try contacting the folks at JW Custom Hats to see if they have any ASM 1107-1 to sell you (it will be at least $2,000 and up). https://www.jwrhats.com/ I am sure that there are other brands back in the golden days of hat making that made machines specific to this task as well but I do not know the model numbers.
There’s an ASM on ebay for $2,600.00 OBO plus shipping. You need to make a lot of hats for that to amortize.