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I modified Max Alekseyev's answer. The result of the following codes agree with the result of the function LongestPerm in the post.

def LongestPermWInSBWInverseSA(W,A,B): 
    t1=StablizerOfTuple(A)
    t2=StablizerOfTuple(sorted(B))
    s=W.simple_reflections()
    #print(t1,t2)
    t3=[]
    for i in t1:
        t3.append(s[i])
    t4=[]
    for i in t2:
        t4.append(s[i])
    t5=find_permutation2(B,sorted(B))
    w=W.one()
    for i in t5:
        w=w*s[i]
    #print(t4,w,t3)
    r2=LongestPermInDoubleCosetWeylGroup(W,t4,w,t3)
    r=r2

    return r

def find_permutation2(L1, L2):
    perm = Word(L1).standard_permutation() / Word(L2).standard_permutation()
    assert [L2[i-1] for i in perm] == L1
    r=perm.reduced_word()
    return r

def LongestPermInDoubleCosetWeylGroup(W,S1,w,S2): 
    g1=W.subgroup(S1)
    g2=W.subgroup(S2)
    winner = W.one()
    for u1 in g1:
        for u2 in g2:
            t1=u1*w*u2
            if t1.length()>winner.length():
                winner=t1
    r=winner

    return r

A=[1,1,2,3,3,4,6]
B=[12,9,12,15,15,14,12]
k = len(A)
W = WeylGroup('A'+str(k-1), prefix = 's')
t1=LongestPermWInSBWInverseSA(W,A,B)
print(t1)

I modified Max Alekseyev's answer. The result of the following codes agree with the result of the function LongestPerm in the post.

def LongestPermWInSBWInverseSA(W,A,B): 
    t1=StablizerOfTuple(A)
    t2=StablizerOfTuple(sorted(B))
    s=W.simple_reflections()
    #print(t1,t2)
    t3=[]
    for i in t1:
        t3.append(s[i])
    t4=[]
    for i in t2:
        t4.append(s[i])
    t5=find_permutation2(B,sorted(B))
    w=W.one()
    for i in t5:
        w=w*s[i]
    #print(t4,w,t3)
    r2=LongestPermInDoubleCosetWeylGroup(W,t4,w,t3)
    r=r2

    return r

def find_permutation2(L1, L2):
    perm = Word(L1).standard_permutation() / Word(L2).standard_permutation()
    assert [L2[i-1] for i in perm] == L1
    r=perm.reduced_word()
    return r

def LongestPermInDoubleCosetWeylGroup(W,S1,w,S2): 
    g1=W.subgroup(S1)
    g2=W.subgroup(S2)
    winner = W.one()
    for u1 in g1:
        for u2 in g2:
            t1=u1*w*u2
            if t1.length()>winner.length():
                winner=t1
    r=winner

    return r

def SymmetricGroupActionOnListSi(i,L): #  w=s_i,  L=[a1,a2,...,an]
    r1=[]
    for j in L:
        r1.append(j)

    t1=r1[i-1]
    r1[i-1]=r1[i]
    r1[i]=t1 

    r=r1

    return r

def StablizerOfTuple(A): # A is a list, weakly increasing
    k = len(A)
    r=[]
    for i in [1..k-1]:
        t1=SymmetricGroupActionOnListSi(i,A)
        if t1==A:
            r.append(i)
    return r

A=[1,1,2,3,3,4,6]
B=[12,9,12,15,15,14,12]
k = len(A)
W = WeylGroup('A'+str(k-1), prefix = 's')
t1=LongestPermWInSBWInverseSA(W,A,B)
print(t1)