Perhaps I did not explain myself well. Tthe idea or concept of mass is "basic" or "fundamental" in the sense that it requires no advanced mathematics to understand. (Most humans have at least an vague idea that objects feel heavy and resist motion.) As I am using it, basic does not mean that the mass of an object is unchangeable.
Perhaps a better word than "basic" would be "axiomatic."
Other basic or fundamental physical concepts are length, time, and (perhaps) force. Most people have an intuitive feel for these concepts; they cannot really be explained in terms of simpler concepts.
BTW, under what circumstances is mass convertible into energy? Einstein's famous mass-energy equation (the one equation everyone seems to have seen) implies not that mass and energy are interconvertible, but rather that the mass of a body is a measure of its energy. Increase the energy, you increase the mass.
That one is apparently what happens in a nuclear reaction of the atom bomb kind. The tremendous power of the bomb is due to conversion of a small amount of mass into a lot of energy. That's what they say, anyway.
As far as basic units are concerned, force is thought of as a product of the mass and the rate of change of the speed of the object. And then speed is a product of space and time. We're not real clear on what time might be, and why it is considered a dimension like height, width, and depth. It gets worse. We say space is 3-dimensional, but it is treated as 4-dimensional when they consider time, and now physicists are thinking in terms of more dimensions, 11 maybe. I don't know, I'm still open to suggestion.