In the 1940s is was pretty common to avoid US import duties by shipping watches as bare movements and assembling them into locally made cases. This watch is an example of that. The movement has the LXW code for import into the US, and the case-back indicates that it was cased in the US by the Longines-Wittnauer company. For this reason, if you get an extract from the archives for the movement, it likely won't have a specific reference, and it will probably verify that it was shipped to the Wittnauer Company. It will of course give you a specific date.
The photo is too blurry for me to get a good look, but the movement appears to have an early shock protection device.
No, the watch is not valuable, and it looks like the dial has probably been repainted. There's no need to authenticate. You can use Longines free service to get an Extract of the Archives. It won't add much value, but it's interesting anyway, and free. You just need to fill out an online form and upload some photos. You don't want the authentication service, just the historical information. Let me know if you have trouble sorting it out online and I can go look and give you a more detailed explanation.
To be totally candid, the value is not much more than the melt value of the 14k gold case.
Looks like a 14-karat gold case. You have bought a watch with a movement that is very much in need of a thorough professional recondition. It looks to be all there, with no serious damage evident, but it sure looks tired. This watch is neither shock resistant nor water resistant. Most people today, consider a watch to be a disposable item, to be used up and turfed when they quit. This one has potential, but should see gentle, respectful wear as an occasional wearer. If it was mine, one of the first things I would do would be to dump that entirely inappropriate high dome crystal, and replace it with a low profile crystal. A low dome crystal would leave lots of room for the hands to clear. The case should be cleaned only, and light polish by hand with a polishing cloth, but not buffed.
You did fine, sorry about my bad math originally (I edited this post). Gold is high right now, $55usd per gram for 24kt. And if you like the watch and want to wear it, then that's more important than the gold value.
One last thing, the strap sales for 135 pounds, would it be smarter to sell the strap, and replace it with a cheaper hirsch just to make my deal a little better? Or keep the strap on the watch?