Depends on how 'slightly damaged' the original box is. If it's just a scuff or a dented corner it probably isn't significant as long as all the accessories (extra straps, spring bar tool, extra springbars), cards, manual, booklets, loupe, paperweight are present. Are you sure you have all the accessories? I don't think the missing pillow is a big deal, plenty of generic ones around that would fit the bill.
I would NOT do a swap from the suitcase to a generic wooden box, they should give you the wooden box and you keep the suitcase.
Yes, there are some relatively recent 1861 based Speedmasters that came in wooden boxes, the two CK2998 LEs and the FOIS came in specific black wooden boxes with the appropriate markings on the inside. The Apollo 8 comes in a glossy black wooden box. And the 44 mm automatic Speedmasters come in wooden boxes.
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