> How would you convince them to abandon their plan to dive into project management and use an existing solution?
I wouldn't. Pushing for "your" solution rather than the right solution is being a fan, not a professional.
I would instead work with them to come up with a list of requirements. Note that that a requirements document is needed in order to do either correctly - to either build or buy, you need to know what features the solution needs to have. Applying a "checkbox" style to the list might be a good idea, to visually emphasize that the right solution is that one that checks off all of these needs.
Then with the requirements list in hand, you look at each option - the existing one, off-the-shelf solutions, and a schedule / quote to build a a custom solution.
If an off-the-shelf solution meets al of the requirements, you show them that - here's the list of 20 things we figured out you need, and this solution checks off all 20 boxes. If no off-the-shelf solution can check off all of the boxes, you ask if any of them can be customized to check off all of the boxes. If not, you must either build custom or revise your requirements.
This process will find the right solution, rather than convincing them to do it your way, against their better judgement. Remember, there's at least 50/50 chance that you're wrong. The other people are just as likely to be right as you are. Listing the requirements as a checklist will answer the question, in a clear, convincing way.
I once asked a couple of friends who'd heard me argue a proposal whether they thought I had done a good job arguing my view. They surprised me when they answered by shrugging and saying "well, you were right. I don't think you did a god or bad job of convincing us, it just became clear that your view is correct." When you present a clear set of facts showing which way is right, you don't need to "convince" anyone to do it "your way", you've simply demonstrated which way is in fact the right way to go.