First of all, it helps to enclose the phrase in quotes when googling:
- "bear the shame" — 607,000
- "bare the shame" — 83,400
Secondly, having looked through the first 10 pages of the Google results for "bare the shame", exposing would not work in most of those contexts at all, it's quite obviously carrying. And quite a few of those contexts feature extremely poor grammar and punctuation.
Lastly, while the British National Corpus has 1 cite for "bear the shame", and the Corpus of Contemporary American English has 10, neither of them has a single cite for "bare the shame".
So, to me, the answer is pretty clear: "bare" is simply a common misspelling of "bear", whether in the context of this expression or elsewhereor elsewhere.