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The answer is no : if you run the following Magma code :

M:=Matrix(Integers(),4,4,[1,0,1,1,0,2,1,2,0,0,5,1,0,0,0,10]);
M:=M+Transpose(M);
L:=LatticeWithGram(M);
H:=GenusRepresentatives(L);
for h in H do
    print "h= lattice with Gram", GramMatrix(h);
    hd:=DualBasisLattice(h);
    MD:=37*LLLGram(GramMatrix(hd));
    print "rescaled dual = lattice with Gram", MD;    
    a,b:=IsIsometric(h,LatticeWithGram(MD));
    print "are isometric : ", a;
    print " ";
end for;

on the online calculator, you obtain the following result :

h= lattice with Gram
[ 2  0  1  1]
[ 0  4  1  2]
[ 1  1 10  1]
[ 1  2  1 20]
rescaled dual = lattice with Gram
[ 2  0 -1 -1]
[ 0  4 -1 -2]
[-1 -1 10  1]
[-1 -2  1 20]
are isometric : true

h= lattice with Gram
[ 4 -1  2  1]
[-1  4 -1  0]
[ 2 -1  6 -2]
[ 1  0 -2 20]
rescaled dual = lattice with Gram
[ 2  1 -1  0]
[ 1  8 -4  1]
[-1 -4 12  2]
[ 0  1  2 10]
are isometric : false

h= lattice with Gram
[ 4  1  1  1]
[ 1  6  3  1]
[ 1  3  8 -1]
[ 1  1 -1 10]
rescaled dual = lattice with Gram
[ 4  1 -1 -1]
[ 1  6 -3 -1]
[-1 -3  8 -1]
[-1 -1 -1 10]
are isometric : true

h= lattice with Gram
[ 2  1  0 -1]
[ 1  8 -1 -4]
[ 0 -1 10 -2]
[-1 -4 -2 12]
rescaled dual = lattice with Gram
[ 4  1  1  0]
[ 1  4  2  1]
[ 1  2  6 -2]
[ 0  1 -2 20]
are isometric : false