Here is an update on my project.
I finally completed the cleaning an reassembly of the watch. It is now running, I have been wearing for a week and it's accurate enough. I do not own a timegrapher so I can't give you an accurate reading.
The missing setting lever spring has been scavenged from a second AS 1187 watch I purchased of eBay.
I also noticed some corrosion on the pallet fork. I did not replace it.
I ordered a new crystal, but somehow never managed to install it. So I put back the old one.
The watch was cleaned using zippo fluid. I have oiled it with Moebius 9010 and D5.
Thank you all for your help and advices.
After waiting a couple of days to receive my Bergeon 080 screwdriver, I managed to strip down the movement.
Since my stem issue, I waited for the power to be completely depleted before attempting to take it apart. Unfortunately upon removing the pallet fork, I noticed that the movement still had some power and the pallet fork got caught into gear train. Exactly what I was not supposed to do. I believe I haven't damaged anything.
Otherwise, the movement is now bare and ready to be cleaned. You'll notice that I left the mainspring in its barrel and the ratchet click hasn't been touched.
I'm planning on cleaning everything with Zippo fluid. I do not have an ultrasonic cleaning machine at the moment.