No, that charger will be capable of charging an 11.1V 5000mAh pack at 4.4A because it's power rating (50W) will only allow you to charge such a pack at that rate. 50W/11.1V = 4.4A, or using my method, 50W/12.6V = 3.97A.
1C on a 5000mAh pack is 5A (5000/1000), so a 50W charger won't charge your pack at 1C.
To charge at 2C, it must be capable of 10A AND have a power rating of over 126W (12.6V x 10A = 126W). Oh, and be 3S capable.
Something like
this charger would fit the bill. I have two of those, and one of the iChargers I mentioned earlier...
EDIT: ah, a ninja edit from you there, you hadn't written that much when I started writing this post.
Yes that charger would fit the bill. Even with a pinch of salt with those ratings it would do. 2C max on your 3S 5000 pack.