All I say here is to my knowledge, I am open to correction.
The order to spend time with a parent is legally binding until a child is 16. Aside from exceptional circumstances, orders on time with the other parent expire at age 16. From that age, a competent child not subject to pronounced welfare concerns. Can make their own decision.
Solicitors are liable to offer advice that involves invoices to you for their services. Having said that, you may be in a situation where your ex has a chance of making mud stick and the solicitor might be protecting you.
How independent are your daughter's views on having supervised time with your ex?
If they are totally independent. What is their basis?
I would think about a couple of things:
1. Explain the possible consequences of not following the order to your daughter. Try to use this as a basis for playing along with the arrangement. At least initially. If people supervising the visits can report that your daughter turned up due to your encouragement. But, things quickly fell apart because she did not want to be there. That is better than not even getting started.
2. Attend family therapy with your daughter and have these conversations with the assistance of an independent third party. This would make a record of your attempts to encourage supervised time, and your daughter's refusal.
I am tempted to say ignore the order, they cannot force a 15.75 year old, she'll be 16 before this gets anywhere.
But, your solicitor's advice and a court order against the wishes of a child this age. Make me cautious that something else is going on here.
I fear the court is not satisfied with your position. If it was, I cannot see how the order of 2 months ago would have come about.