The parts that are visible look gen, but the dial was repainted at some point. If you are considering a purchase, I would ask seller to send good, clear macros of the movement & engravings between each of the lugs. That will give you a better idea of what it is.
Avoid reps with "gold" (it fades or rubs off), "jewels" (always look fake) or alot of complications (features that do not quite 'look' right).
The more details on a rep, the more likely it looks like a rep; the simpler (fewer features), the more likely it will pass as genuine (gen). & read Toad's guide.
Other than gens, I have never seen rep/aftermarket 118138 straps available. Assuming you do not want to buy the gen strap, why not just pick up a cheap rep DD & just use the strap?
A typical case of someone (the other guy) trying to do you a favor, but, in dragging his feet, ends up causing more confusion than help.
Considering the part itself did not cost you anything, I would either give it another day or 2 or source another part from a more reliable source like a watch parts seller.
I am unfamiliar with an SOA44, but check the threads on both the tube & inside the crown to see if they are cross-threaded or otherwise damaged.
If both are ok, then either the stem may have become dislodged or the keyless works are misaligned (you will need to remove the movement from the case & the dial/handset to check this). If unsure or you lack the requisite skills/tools, this is definitely something to let a pro take care of.
I think I would wait until you have unpacked everything before losing your grip. Moving, as you correctly stated, is very unsettling & you probably had a dozen things going through your mind when you packed your watches (without thinking) into what seemed to be a logical place. Unless you transported your watch cases loosely, so handymen would be handling them directly, I would not worry about them. I am sure they will turn up.
The general axiom is - you break it, you own it. Considering the condition, unless the damage was accidental (& you feel like doing the customer a favor), I would happily offer a replacement at full price or the option to smear away.