We let her have the base, giving it to her at naptime and bedtime just like we would have done before. She said a lot of 'Uh Oh!' and 'Oh No...' So we just told her binkie was all done, agreed with her every time she brought it to us, 'yup, binkie uh oh, it's all gone, it's all done' and tried not to make a big fuss about it. We let her take the base to bed with her, which was a little comical when she tried to put it in her mouth. She did cry a little the first few naps and nights. Nothing too horrible, just that complaining cry that you know isn't serious.
After about a week of sleeping with the base she stopped looking for it in her bed, so I just took it out.
It's been about a month and a half now, and she's doing just fine. Sometimes I wish we still had one when she's being really cranky, but mostly I'm happy she's done with it!