My guess why Meredith seems so antagonistic towards Le Fay is simply jealousy / envy.
Despite her (Meredith) status as member of royalty, the Pendragon siblings are more prominent, both in the occult world and within the nobility - Arthur and Le Fay both being comrades to Vali, the strongest Hakuryokuo(With Albion being a symbol of England no less), with Le Fay even being contracted to the most well known entity in the supernatural world in modern times, the Sekiryuutei Issei.
Considering that, in some tales at least, the wielder of Excalibur(Colbrand in DxD) have a right to the throne in addition to their fame / prominence, I can see both her and members of the royalty being concerned for the future. Political among nobility is always present, be it to replace a regime or simple political gain. (If I'm not mistake, it was also mentioned that the Pendragon family were a prominent family in DxD). Even if she did not originally look at things from this perspective, I can see the influence other adults would have on her.
It was mentioned that her existence was hidden by the Royal Family, so it could be argued that she lived a very sheltered life, hidden by all but a few in the know - the complete opposite of the lives both Le Fay and Arthur have lived thus far.
All of this could have have potentially combined to create the animosity she currently feels for Le Fay.
... Or, you know. Boobs
As for her Longinus's ability, if I'm not mistaken, it is the thorn of crowns, yes? If so, I can see it having the ability to "Control". In application, while it may be able to control others to a certain degree, being a Longinus I can see it more so having the concept of control. Kinda like Asuka from Mondaji.