The one method without holy weapons I know works for sure is using berserk on whichever warlord has the Hel tome. FE4 AI makes it so he attacks Julius, reducing him to 1 hp, so anything can finish him off. Of course, the chance of that happening is ridiculously low barring RNG manipulation. The biggest problem with killing Julius without holy weapons is that he has wrath, so units without awareness would be slaughtered when he went under half health. I think that due to how Lopt reduces stats, any unit would only deal one damage.
In terms of bringing Julius down, you could give the sword twins hero swords (they inherit awareness from Aira), Celice could use the hero lance, via pairing Aideen with Alec both Lester and Lana would have awareness, so that would be a hero bow and a tome attack. Julia can also chip since she has awareness too. Altogether, including the counter attack on enemy phase, that would be 24 damage a turn, 14 of which would be healed by Julius, so it would be doable, just take a while and require a lot of healing. With good dancer + rescue staff use you could probably get in even more damage per turn, but I can't be bothered to work out the optimal way to do that.