Flight Instructor reported an ATC assigned vector contributed to a NMAC and the need to conduct an evasive maneuver to avoid a collision while in the pattern at an airport with an overworked contract tower.

Narrative

My student and I were doing left patterns using Runway XX at ZZZ. On late downwind; we were told to follow traffic at 12 o'clock. We couldn't find that traffic and were told to make a 30-degree turn to the right. I informed Tower that such a turn would put us into the ZZZ1 bravo surface area in a few moments. Tower argued back and told us to just do the turn. We complied and as we were turning right; we spotted a white wing passing right below us. The traffic was; in fact; not at 12 o'clock but actually closer to 2-3 o'clock. Upon spotting the traffic; I grabbed the controls from my Student and told the Tower about the traffic. They told us to follow that traffic; #2; cleared for the option. I read that back; but informed Tower that we were continuing the right turn into a 270 back into the base leg as we did not have enough room to go left back to the base without clipping ZZZ1's surface. Upon hearing of our 270; I heard the Controller give an annoyed sigh on frequency and tell the traffic behind us to do a 360 for spacing. As we came out of the 270; we reacquired the original traffic that almost hit us. The spacing was much better now and we continued to land uneventfully. The ZZZ Tower has been critically understaffed for a long time. It is a contract Tower. They routinely have to work through long hours with no breaks and are often stressed and overwhelmed as a result. Mistakes and errors are extremely common now; including the spacing error described in this report. ZZZ needs a better Tower.