wire brushing is my first choice. either by hand or with a "powered wheel" in a drill. some sturdy leather gloves and safety glasses are a must.
with coca cola and than with a wire brush
Old rust will be there forever if all you do is clan the surface rot off.
It's been many years since i did archaeology excavating and I've forgotten a lot, but I suggest you browse for archeology sites that can lead you to the rust removing info you need.The rust is deep down and will come back eventually if you just clean off the surface. It could be that you won't keep the doggie bed long and it's just another short term item, of which we all have too many I sometimes think, and in that case cleaning off surface rust with anything that will take it off will be okay. I remember that in the 60s we used co-cola to take the floor wax off cork tiles......so could be it'll deal with this issue, too..... don't know if this helps much, but good luck. PS Any university that has an archeological dept should be able to connect you with a 'rust expert.'