I don't think there'd be a problem, though like Stopdog said, I'd expect a therapist to ask you why you were carring a chair around with., because it's not really typical. But, it's not a problem, just a little unusual.

Do you think that your T might have access to other, more comfortable chairs? I'm just thinking if you can ask her to bring in another chair, it might be less work than having to carry one back and forth every week.

Good luck. Not having my feet touch the floor at all would bother me too... I think I'd feel a bit ungrounded and vulnerable.