Hell, I even have the nail to justify it!
The Einzberg came to the logical conclusion that their choice of Servant during the 4th War was perfectly good, since she won, it was Kerry's fault that everything went to shit. So really, why bother with another Servant ? So they go and acquire Arthuria's spear somehow and summon her again, tough she's a Lancer this time.
As a side note, this mean that Cu automatically get shuffled to the Berserker, since the Lancer class is already taken, on summoning. Kirei has a Bad Time of it.
Fast forward to the proper beginning of the 5th War, Shirou's heart get blown up somehow, maybe by Arthuria herself on Illya's order using a Command Seal, get chased, summon.
Problem is, Arthuria already here! You can't summon duplicate! That's not done! So the Grail has to search for another Heroic Spirit with an affinity for Avalon. And would you look at that ? Right there, there's one who just happen to be a hommonculus built specifically for the purpose of being as close to perfect physical and spiritual clone of Arthuria as you can get! Why, Morganna went out of her way to make sure that Mordred would be fully capable of using Avalon and Excalibur by fooling their DRM into thinking that she's actually Arthuria.
Enter Mordred, right in front of a Seal Commanded Arthuria. Let the epic grudge match begin!