Mitford Mathews, A Dictionary of Americanisms on Historical Principles (1951) has this entry for "foot washing Baptist":
foot, n. ... 3. In comb[ination]s: ... (13) [foot] washing, an occasion upon which Primitive Baptists wash each other's feet as a religious memorial; (14) [foot] washing Baptist, a member of any one of several sects of Baptists who practice foot washing;
Also of note is Mathews's entry for "Primitive Baptist":
Primitive Baptist, a Hard Shell or Old School Baptist. Also attrib. [First citation is to Polly Peablossom (1851)].
And for "Hard-shell Baptist":
Hard-Shell Baptist, a member of the Primitive Baptist Church, or Old-School or Antimissionary Baptist. [First citation is to Knickerbocker magazine (1845).]
Hard-shell has the transferred sense of "A severe or strait-laced person," according to Mathews.
One early Google Books match for "foot-washing Baptists" is from Annual Session of the Baptist Congress (1899), in an address to the congress by Rev. Emory Hunt, of Toledo, Ohio:
In the records of the Miami Baptist Association, the first formed in the northwest territory, we find that in 1807 a query came up from the Union church on Indian Creek, "whether the washing of saints' feet be an example left on record for the professed followers of Christ to be continued in his church." The association laid it over for one year and then replied, "we consider every church independent, and if the church on Indian Creek, or any other, agree among themselves on this point it will not affect their fellowship with their sister churches." We may doubt whether a foot-washing Baptist church would be made comfortable in a Baptist association to-day.
It thus appears that "foot-washing Baptists" were a category of Baptists who believed in emulating the biblical ritual of washing the feet of holy persons in some of their ceremonies. Further, when Miss Maudie says in To Kill a Mockingbird that she doesn't have as hard a shell as that, she simply means that her religious beliefs are not so straitlaced and severe as those of Hard-Shell or Primitive Baptists (who evidently consider foot washing to be an appropriate religious ritual).