JAEGER LE COULTRE caliber P478 Identification

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Hello everyone
My name is Doreen and I’m new to this forum.
I’m very happy I found this forum as I’m impressed with all the useful information and with all the interesting posts and images.
I recently purchased a vintage JAEGER LE COULTRE caliber P478, movement serial number 370849, case reference number 310477, case dimensions without the crown 33.5mm with blue hands.
Dial has the SWISS on the bottom along with a beautiful vintage patina; some dirt and dust.
I will attach some images for your reference and hopefully I will get some help identifying this beautiful watch. I don’t have many knowledge about this particular brand”JLC”.
I’m interested to know as many information as possible like: year, if the dial, hands, crown are all original and belong to this particular model.
I’m also wondering if this particular model is a desirable one to add to a small watch colección.
I don’t even know if I paid to much for this one $550. I’ve seen this model selling for about $2000, of course that doesn’t mean that is the real value of the watch.
The movement is very dirty; I have to clean it and oiled soon. Also I intend to “clean”a little bit the dirt and dust from the dial. Any suggestions on how I should address this? I’m thinking on trying with a little bit of warm water and some q-tips and go very gently over the dirt and dust; perhaps just tapping slowly not rubbing. I will definitely not touch the old lume from the numbers or the hands.
The Case looks not polished for my unprofessional eye also the crown looks original but again, not sure.
I tried to add a few images unfortunately I don’t know why they are not downloading.

Many many thanks to all who can provide some information...very much appreciated.

[zafbasha]

Hi, I don't advise cleaning the dial. The odds of making it worse or removing print is very high.

[brutus]

Thank you for your response.
I just finished cleaning the dial and I’ve done a little bit of restoration too on the black painted “dots” on top of the numbers. If you take a look on the first image, that is the one “before” and the last on is”after”. The dial had a scratch on the black markers”dots” along the half of the dial from number 12 all the way to 5-6. Perhaps from the minute hands that was probably touching the dial.
I will try to download some images...not sure if is working.

I’m quite pleased with the results. Also I’ve finished to clean the hands too. Pleased with that result too. Tomorrow I will start working on the movement. 

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This is the dial cleaned.

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Any idea about the year on this movement based on the serial number?
If I have to guess I will say 1945-46, but I’m not quite convinced.
Many thanks

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This is the watch.

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Watch

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Back cap

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S

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