I used to work in University IT and registration time is the worst. The systems never handle the load well and everyone freaks out. I think your work might anger the sysadmins b/c of the increased load.
Using this example, couldn't someone (University IT) write an SQL script to dump out the current seat count/capacity for every "current" class and throw it on a web server somewhere? Assuming that the underlying database schema isn't completely insane, that would probably be about a half-day of effort. The student writing the scraper could then point it at the output of this script.
With a half-day of effort, you've reduced the load on the application, you've provided a service that the students probably want, and best of all you've got someone else to do most of the hard work.
Students rank their course preferences in an order and submit it. For each course, free seats are first allocated to the students who ranked it as their first choice, then what is left to the students who ranked it as a second choice, and so on until the round n and there are less seats available than people who ranked the course as their nth choice: the remaining seats are allocated by a lot to them. (For a very popular course, they would have to start with the random selection right away on the round n=1).
However, instead of enrolling the lucky students right away, everyone gets notified about which classes they have been tentatively admitted, and they must confirm their attendance before they can take their seat. If they don't confirm in a timely manner, they lose the opportunity and the free seats are propagated to students left in the waiting list until someone accepts (again using random lot).
The one problem I can think of: the process (especially the propagation of rejected offers in the waiting list) may be too long-winded to be practical. So maybe skip the confirmation part: there's a set maximum number of courses you can take, M, and slots are filled until you're enrolled to maximum of M courses.
But in any case, if the overbooking is enough of a problem so that you are looking at methods like these, the real problem is that have too many students compared to the teaching resources. All other attempts can only mitigate the symptoms of the root cause.
So maybe staff should choose limit the amount of students eligible to take some very popular courses by an entrance exam, or imposing certain mandatory minimal GPA requirement in the prerequisite courses. This option should be especially considered if the attrition rate is high (there's a problem of students taking much-coveted seats in the class but who fail to show up or otherwise don't put in enough effort to pass). Usually that would have been done on the admission level already, though.
"All students are given an equal amount of points per semester to bid for modules... The allocation of modules is based on the lowest successful bid points against the last available quota for the module at the end of each bidding round. If supply (module quota) exceeds demand (number of bidders) for a module for any bidding round, the lowest successful bid will be 1 bid point. If there is a tie in the lowest successful bid points, the outcome will be based on first-come-first-served. Unsuccessful bidders will be fully refunded. Any unused bid points after each round will be carried over to the next bidding round or to the next semester at the end of the registration exercise."