A self-taught industrial designer, I don't think that's out of the question if your prohibitions to get a degree are that high.
The BA in interior design gives you a little bit of an advantage as well.
If this is what you love, go for it.
You do need to realize you will need to work extra hard - you have to make sure that you establish some contacts with professional experience to guide you, get feedback on your projects from professionals early-on, you have to make sure that you can learn in a hands-on way, using books and computers only as supporting tools for the actual product development, you have to make sure that you follow design methodologies and are in line with the practices of the industry etc. So you would basically have to design your own education first. There is lots of literature available online to start off with.
And then still you need to realize that people with ID degrees will in most cases be picked over you when applying for jobs, unless you prove to be exceptional in certain areas.