I don't know how well this fits your usecase, but Proxigram is a selfhostable website with which you can access Instagram anonymously. I don't think it supports following accounts in any way though.
You can try it out on one of the public instances.
I did also try making an android app for it, but that app is barely even able to display posts, and as I realized that I just don't use Instagram, I didn't really bother continuing it.
If anyone here is in the mood to make that app happen, feel free to use my humble beginnings as a starting point.