I think it’ll be okay if a simple icon is displayed on the Quests button, similar to the heart icon used to indicate that we have received a Cheer from our friends.
Perhaps can display a number with a red Trial Energy icon at the top right corner of the Quests button on the field map screen and put a number, from 0 to 5, to indicate the amount of Trial Energy available.
This way, it should not clutter up the main screen further.
The Trial Quests button on the Quests screen should also have the same marker so that the player can “follow the breadcrumbs”, so to speak.