Based on ti7's very helpful answer I've slightly modified their solution so that the replacement Pow(a,b) -> a happens for any type Binary symbol, regardless of a being inside an IndexedBase or just a Symbol. After all, the identity x**n==x holds for either.

from sympy import *

class Binary(Symbol):
    '''Empty class for tagging variables as binary'''
    pass

def simplify_binary_powers(expr):
    '''
    Remove exponents of binary variables by replacing Pow(type(Binary),b) 
    for type(Binary).
    '''
    a = Wild("a", properties=[lambda a: a.atoms(Binary)])
    b = Wild("b", properties=[lambda b: isinstance(b, Number)])
    return expr.replace(Pow(a, b), lambda a, b: a)

Works for IndexedBase:

x = IndexedBase(Binary("x"), integer=True)
expr = x[0,0]**3 + x[1,0]**2 + x[2,0]
print(simplify_binary_powers(expr))

Output: x[0, 0] + x[1, 0] + x[2, 0]

and also works for a single Binary variable:

y = Binary('y')
expr2 = y**3
print(simplify_binary_powers(expr2))

Output: y