I get $$\left\{\frac{2}{3},\frac{2}{3},\frac{4}{9},\frac{35}{81},\frac{94}{243},\frac{275}{729}
   ,\frac{263}{729},\frac{2267}{6561}\right\}$$

for the monic polynomials of degree 1 to 8, using Mathematica:

    f[a_, b_] := a x^(b - 1)
    
    PolysOfDegree[n_] := Map[(x^n + Plus @@ MapIndexed[f, #] // First) &, 
                             Table[IntegerDigits[i, 3, n] - 1, {i, 0, 3^n - 1}]]
    
    TestFactors[n_] := Table[FactorList[x^i - 1], {i, 1, 2 n + 2}]
                       // Flatten // Union // Rest
    
    HasRootOfUnityAsRoot[poly_] := Or @@ Map[ PolynomialMod[poly , #] === 0 &,
                                              TestFactors[Exponent[poly, x]]]
    
    Prob[n_] := Count[Map[HasRootOfUnityAsRoot, PolysOfDegree[n]], True]/3^n

    Table[Prob[n], {n,1,8}]
 

I've enumerated the polynomials of degree $n$, and enumerated the characteristic polynomials of roots of unity of degree $2n+2$.  Then it's just a matter of testing which are divisible by which.