No, I'm pretty sure that's actually what BlackHadou intends if Illya wins. He's a huge Mordred fan, and Mordred meets the qualifications for Berserker. Mordred made a deal with Alaya, making her a counter guardian, which can be summoned by the Grail. Shirou's catalyst is Excalibur's sheath, Avalon. Meaning it can really only grab Arturia or EMIYA. Except Shirou summons last, so EMIYA is already summoned due to Rin accidentally using her necklace as a catalyst, and Arturia will already be summoned by Illya. Which means the only hero left who is vaguely connected with Excalibur and meets the qualifications for Berserker is Mordred. ...And Lancelot, now that I think about it.
However, Mordred makes for a more dramatic story than Lancelot, and again, BlackHadou is a big Mordred fan.