How long have you had it? "Mini" refers to the size of the bloom, not the size of the plant. I have "minis" that are over 3' tall. It is probably own-root, so that's not a sucker, it is a basal break from the roots. Yes, roses do that, minis too. Sorry, but it looks like your "baby" has grown up. You can try to keep it pruned back, but it probably won't be happy.

I planted Baby Cecile Brunner as a border plant but had to move it to the back of the rose bed. It's now about 6 feet tall. It grows upright, not sprawly at all. Perhaps it has decided to be a climber and not a mini. It doesn't look like its parent Cecile Brunner, Climbing but must have some of its tendencies. Can't you move your rose and see what happens?

Baby Cécile Brunner breeding: Cécile Brunner, Cl. (Climbing, Hosp 1894) x Fairy Princess (Miniature, Moore, 1955)

Here is a link that might be useful: Baby Cecile Brunner at HMF