In The Handmaid's Tale, why would the Baptists rebel against a fundamentalist regime?

While the text clearly indicates that the Baptists are rebelling against the Republic of Gilead, there are some very significant clues that the theology of the Republic is, in fact, based less on the Christian Gospels than on the Old Testament.  

First is Atwood's epigraph quoting Genesis. In the reference, she provides the justification for Handmaids. It is the text referencing Sarai's giving to Abram (not Sarah/Abraham yet, so the Hebrews were not yet...

While the text clearly indicates that the Baptists are rebelling against the Republic of Gilead, there are some very significant clues that the theology of the Republic is, in fact, based less on the Christian Gospels than on the Old Testament.  

First is Atwood's epigraph quoting Genesis. In the reference, she provides the justification for Handmaids. It is the text referencing Sarai's giving to Abram (not Sarah/Abraham yet, so the Hebrews were not yet extant) her handmaid Hagar to bear him an heir.  

Second, we read about Marthas, rather than any reference either to Mary the mother of Jesus or Mary the sister of Martha and Lazarus.  The last, in particular, is significant.  Martha, who rushes around doing work rather than sitting at the feet of Jesus to listen to his teachings, is the woman who is chosen as the namesake of the serving class of women.

I would also caution you in equating Baptists with religious fundamentalism.