I know that one can set a maximum size (in pixels) for images uploaded through the CP, but is there any way to set a minimum size? (Ideally without buying a complete image manipulation suite like CE Images, since this is a low budget site and this is the only feature I'd need).
I don't have a free solution but Photo Frame will do what you're after. https://objectivehtml.com/photo-frame/settings
This might depend on how you are going to use this. File upload with a Channel Form or via a WYSIWYG editor. However both these examples don't have a straight forward configuration but you could write a conditional with a minimum file size.
CE-image will perform practically any image manipulation you can think of, but it also cannot set an minimum upload size for the original file.