Store in each classroom a custom-made rack for the laptops (with at least one laptop per seat plus backups). Distribute the laptops at the start of exams and collect them at the end.

The rack charges the laptops, streamlines distributing/collecting tests, prevents tampering, and reports defects. A professor uploads the test to their LMS and specifies the exam time-frame and location. Before the exam, the LMS transfers the test to the correct rack, which saves it to the laptops. After the exam, the rack loads the student responses and transfers them to the LMS for the correct class. Each laptop has a light under it whose color indicates whether it's in standby, waiting to be distributed (shortly before exam start), has a submitted test, or is malfunctioning (the rack periodically pings laptops in storage to verify they still work). Multiple professors can queue exams to the same location, in different time-frames, and the LMS and rack know when to prepare each test on the laptops.

The hard part is to implement this well. The rack's hardware and software must be reliable; nonetheless it will fail (hardware breaks), and exams get moved and rescheduled, so there must be a way to transfer tests to another classroom and time-slot. The laptops can have slightly less reliability, since there are backups; but failures can't be clustered, and if a laptop is transferred from another rack during an exam, it should download the destination rack's test and become active (so if one rack's backup laptops run out, laptops can be borrowed from other classrooms). The laptops must periodically backup in-progress tests using wifi or bluetooth, and if a laptop breaks during the exam, the student can resume their progress on another. Tests must be downloaded well before exam time, in case there are problems (laptops should be able to store and queue multiple tests). Laptops must handle exams that start late (up to the official exam end time) and end late (including overlapping the next exam's start time, in which case the next exam is loaded when the laptop is put back into the rack). The rack must absolutely not indicate that a laptop is ready with an exam if it's not; and (since tests are downloaded before exam time) nothing unique should happen at exam start time (to reduce last-minute surprises, and let a professor who isn't convinced take out a laptop and check that it's functional 15 minutes before). Last but certainly not least, the UI to submit tests and grade responses should be intuitive, simple (not overwhelming) yet powerful (can handle edge-cases like rescheduled tests, randomized tests, accommodations; or be extensible enough to support these features).

Despite all the above requirements I actually think such a product is feasible. Unfortunately, with firsthand experience of typical EdTech and academic bureaucracy (and I'm not a professor so I don't know the worst of it), I'm skeptical it would be adequate quality unless the company designing it is uniquely motivated and capable.