Hi debouvier, welcome to the forum.
As you suspect there are a number of options here to go at and a number of techniques. Clipping Paths, cloning and some degree of mirroring and the like, all of these are pretty straightforward and don't strain the PS muscles too much.
If there is a strong weave to the fabric and you wish to keep the pattern, whilst removing creases, I would go with a split of the frequencies and more to the bandpass. If you look up frequency separation and bandpass it should put you in the right ballpark. If there is only a little texture or none, then I would go with content aware fill or clone/heal.
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