Student Pilot conducting a practice approach reported the TRACON issued a late frequency change to Tower resulting in a Class D airspace incursion. Tower instructed the student to extend their circling maneuver which placed them in unsafe proximity to terrain requiring a climb to avoid.

Narrative

When heading into Prescott on an IFR plan; I called Phoenix Approach and requested an instrument approach. They said they could not do an actual approach but could do a practice approach instead if we canceled IFR. I canceled IFR and was cleared for the practice approach into the airspace. We were beginning to get close to the airspace so I asked my instrument instructor if we could go in. We debated it for a bit and I said that we couldn't go in because of a lack of two way radio communication and we decided to turn away. My instructor then told me to turn back to the approach course and head inside the airspace. We asked approach if we could switch frequencies; and they said that we were approved for frequency change and apologized for the late transfer. We were already inside of class D airspace at the time; and then contacted Tower. Tower told us to continue on the approach and circle to a runway. When circling; they told us to extend our downwind. We were at circling minimums; and if we kept at those minimums we would have been sent straight into a mountain. We had to climb to avoid a collision. The transfer issue could have been avoided if the frequency had been less congested. We weren't able to ask ATC to switch earlier because of communication saturation. The issue with being told to fly into terrain could have been avoided if ATC was more aware of the circling altitude we're required to fly. The Tower was extremely busy so they may not have been paying attention to us.