Famitsu’s new online advertorial (may be NSFW) for Luminous Arc 3 sheds a little more light on the dating-sim aspect and the bathhouse beneath the magic academy Refi attends, which looks like it will be a major setting for character events in addition to the school cafeteria.
In addition to being able to "relieve the weariness of battle" (*cough cough?*) in the bath, Refi and his toweled female teammates have event sequences at a public bathhouse. You can pay a fee to increase your maximum HP at the bath, too.
The bathhouse isn’t the only place where you can pay money for some useful benefits. In the academy library, you can pay a fee to obtain confidential information about your peers. This will allow you to engage in more interesting and intimate conversations with them. According to Famitsu, Imageepoch have put special care into the sub-scenarios, which complement the main story.
There’s also mention of how the dating-sim aspect affects battles. It seems like getting intimate enough with a certain character will give you a Zodiac Card, which will enable you to perform a powerful attack known as a Unison Strike when that character is adjacent to you in battle. Each character’s Zodiac Card seems to be different, which could make for some interesting battles.