First off, there’s no question that Rumi was interested in knowing God better through direct mystical experience. So he was a mystic. His example of life and his beliefs are not particularly useful to us as Christians, but it’s silly to say it didn’t happen at all. Catholicism does not deny that God occasionally speaks to pagans or to those living in heresy. God is always calling everyone to know Him better, to learn the truth about Him from His revelation. People often accept God’s invitation, and then turn back at the door before entering the Church. This is sad, but it doesn’t mean God’s dealings with them never existed.
The main reason to read Rumi is not that he was a mystic to be imitated, but that he wrote good mystical poetry. Rumi’s poetry is part of the heritage of all educated people. Just as pagan poetry was good enough to be quoted by St. Paul in his Epistles, and just as Donne’s poetry is worth reading by Catholics even though he never joined up with us, so it is useful and interesting to read Rumi’s poetry. As many early Christians have explained in detail, the great art of non-Christian cultures belongs to Christians by right, and should be studied by them. “Whatever is good, whatever is true” belongs to Goodness Himself and Truth Himself, and we are part of Him. Spiritually speaking, great art is “the spoils of the Egyptians.”
As with the Song of Songs and a lot of the medieval and Spanish mystical poets in Europe, Rumi’s poetry is about seeing God as a sort of invincible romantic lover, and the human soul as a nervous girl. If this kind of stuff helps you, have fun. If it weirds you out, don’t read the man.
