The first Hurricane I can find coming down due to enemy action near Verdun is a 1 Squadron machine, L1590, on 23rd November 1939.
'Hit in engine by return fire from Do 17 P of 4.(F)/122 engaged over Bar le Duc, and belly landed near Moiremont, west of Verdun, 11.15 am. Flying Officer C.D. Palmer unhurt. Aircraft write off.'
This is the only one that fits the bill for 1939, though Moiremont is about 20 miles from Verdun..
The damage to that propeller tip is not consistent with a power on wheels up landing in which propellers of this type are usually smashed to pieces. That doesn't mean that it couldn't have survived a dead stick landing, being one of the blades that did not strike the ground.
May 10th 1940 was the opening day of the German offensive in the West, and there was therefore a lot of action that day and the British Air Forces in France ( BAFF), both the Advanced Air Striking Force (AASF) and the Air Component of the BEF suffered substantial losses. I just don't see how any of these can fit the description of "first RAF fighter shot down by the Germans in France", because many had already been lost, or, more obviously the year 1939.
I think you will struggle to nail down a specific aircraft, given the information you have, but this doesn't detract from the fact that you've got a very nice item with an interesting, if slightly vague history
If you feel that the May 1940 date is most likely, then Hurricanes were operated as part of the AASF by Nos.1 and 73 Squadrons already in France, joined by Nos. 501, 17 and 242 from the UK.
Hurricanes of the Air Component BEF came from Nos. 85 and 87 Squadrons, reinforced respectively by A Flight 242 Squadron and A Flight 111 Squadron.
That's a lot to look through!
I would try to get your Godfather's service record. For example knowing where he was and when would narrow the search down considerably. Nearly every serviceman in France is now thought of as being with the BEF (not least since Dunkirk) but of course many were attached to other organisations as you can see from the RAF's Order of battle on 10th May.
Cheers
Steve