Hmm maybe you should not really focus on awakening your LoLs then? The 4* shards drop from LoL 13+14 and for the next awakenings you must be able to grind them and for the final awakening you need to grind LoL15!

There are some threads for the LoL Dungeon here, e.g. from Bakura. There you can get some ideas what to expect and which teams might work against your opponents there. Do you have Charlotte and Viper talis? They would help you a lot for the higher LoL levels!

Good luck!!