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][1], 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


  [1]: http://magma.maths.usyd.edu.au/calc/