Oh boy. A fair but possibly difficult question to answer.
I have been using Linux since 2005. I have also been a car owner since 1974. I will be honest. I know more about how the car works than I do Linux. In other words, I am no expert, but based on experience I will try to answer...
First: Minimal distributions are great for what they can do. But they are minimal for a reason. They are small because users are either looking for speed or the resources of the host machine is limited. ie: Memory, Graphics or CPU etc. I have nothing against 'minimal', for "every day" use, they will be enough for most users. But programmers and the like are not "most" users, are they? lol
Here is a simplistic question. As a Linux user, I could also ask, "What is needed to run QB64 on a minimal version of Windows?". The answer would probably be, "Windows does not make 'minimal' versions."
In my opinion, "full" distributions of Linux, would give you the best "out of the box" experience. Pretty much all you need is already there. Sure, QB64 install requires specific libraries, but same is also said for Windows.
Linux's greatest strengths are that it is open source, big community and customisation. These are also its greatest weaknesses... "Minimal", and such like distributions, are "created" for a multitude of reasons... This would be the root cause for "distro hopping"... looking for the "right" distribution to suit your needs. To be honest, you may not find it, because they were 'not' made to suit 'your' needs.
My suggestion, so as to eliminate your "minimal running" query would be, stick to a "major" distribution... Check out distrowatch.com for a short list of distributions.
I apologise for not answering your specific needs... It's just that I do not know.
All I know is that, on Widows, QB64 installs and runs pretty much "out of the box". The same is said, in my case, for Linux mint. It installed and is still running fine.
Like the average car owner, I will eventually learn the basics of operating the car, but will probably never really understand why it 'ticks'...
I suppose I should have started with, "Why a minimal distribution?"
I hope I have not made your situation any worse...