You asked for advice, so I'll add mine.
You have bought a less common early 1937 Luger with the rust blued / strawed finish and the frame profile from before the "hump" was milled. There are fewer of these than the later ones with humped frame and salt bluing. It appears to be a matching pistol with matching magazine (even though there is a die strike bounce on the "4").
It is worth more than the $2,200 you paid for it with the holster.
Break a numbered part and value drops 35-40% or more. That is the risk you take by shooting it. Even if you can match up a part (which is difficult), it will never be historically matching again.
I personally would not take that risk. It could be less expensive to buy one that is mismatched, or that has finish conditions as a shooter and then take it and shoot it as often as you like.
At one point, tired of the question, I cynically posted in the forum that I should change my advice to encourage people to shoot them so that the rest of the matching ones would rise in value, but I got heartily criticized for that attitude, so I've changed my opinion to just discourage people from doing what you're planning.
Of course, it's yours, and your results may vary... The Mauser steel's metallurgy is the most modern of all the steel used in German military production.
The "Broken Parts Survey" starts at the bottom of page 37 of the August, 2017 version of the FAQ Document. Just search the PDF for the word "broken" and you'll find it, or click on the item in the index.
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 Igitur si vis pacem, para bellum -
- Therefore if you want peace, prepare for war.
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