How should I structure exams when individual problems could use up the entire class time?
The most time-consuming and least information-bearing step (at least for the purpose of evaluating understanding of the material) is 4. In this situation I've often just given questions asking the students to do 1-3 and explicitly told them to not go through the calculation. It's better if you prep them for this by giving such a question on a quiz first since many students tend to go on autopilot after a couple of decades of being trained to come up with an answer.
Split up the question into subsections that, in total, are the initial large question. Turn these into individual questions, in-place of the expected components of the large question.
Identifying the correct differential equations and boundary conditions to solve.
Making the correct assumptions and simplifications.
Identifying the correct solution strategy.
Implementing the strategy and solving the problem.
Wait. Do you want us to tell you how to test with a single large question, or how to make a test that would "evaluate mastery of the breadth of the material"?