I hear what you are saying. When my twins were born I was going through 5 rolls of film a week. Shortly after I purchased a Canon Elfph Point and Shoot camera for my wife that could easly be carried in her purse or the diaper bag. Shortly after that I got a DSLR. Mirrorless was not really a thing back then.
Are you set on Mirrorless. At this time buying a camera and having a low budget are not really in sync with mirrorless cameras. Mirrorless cameras have the advantage with fast focus speeds and tend to be smaller. But that is really the large extent of their advantage.
A significant part of having a camera with interchangeable lenses is just that. the ability to change lenses. At this time mirrorless lenses cost significantly more, as well as the camera. I generally tell peopel who are looking for a higher end camera to look at the accessories and lenses they want to buy to go with the camera.
However, I would suggest looking at the Sony A6000. One of my older daughters has a prior version and it is pretty impressive. But she does lament the cost of lenses.
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