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Here are some heuristics :

Edit Feb. 21 : Let $p$ be a prime, congruent to $3$ mod $4$. Then there is such a genus on the $\mathbf Q$-quadratic space $[1,1,p,p]$. Here is the number of fixed (resp exchanged) lattices:

Edit Feb. 21 : Here are some heuristics (in each case, there is a quadratic space $V$ on which some lattices $L$ with $q(L)\subset \mathbf Z$ will furnish Gram matrices $A$ such as required in the OP (recall that in that case, the Gram matrix is associated to the bilinear form $x.y=q(x+y)-q(x)-q(y)$, in particular it has even diagonal entries).

Let $p$ be a prime, congruent to $3$ mod $4$. Then there is such a genus on the $\mathbf Q$-quadratic space $[1,1,p,p]$. Here is the number of fixed (resp exchanged) lattices:

Edit : Let $X_p$ be the set of isometry classes of 4-dimensional positive definite lattices satisfying the property $L^\sharp/L\simeq (\mathbf Z/p)^2$. The set $X_n$ is stable under the involution $\tau : L\mapsto pL^\sharp$.

The question seems to ask whether or not $X_p^\tau$ is non-empty. Will Jagy has given evidences for a positive answer. Nevertheless, one might ask whether the same holds when one restricts the question to a given genus (genera are also stable under $\tau$).

Let $p$ be a prime, congruent to $3$ mod $4$. Then there is such a genus on the $\mathbf Q$-quadratic space $[1,1,p,p]$. Here is the number of fixed (resp exchanged) lattices:

Original post :

Let $X_p$ be the set of isometry classes of 4-dimensional positive definite lattices satisfying the property $L^\sharp/L\simeq (\mathbf Z/p)^2$. The set $X_n$ is stable under the involution $\tau : L\mapsto pL^\sharp$.

The question seems to ask whether or not $X_p^\tau$ contains an even lattice. Will Jagy has given evidences for a positive answer. Nevertheless, one might ask whether the same holds when one restricts the question to a given genus (genera are also stable under $\tau$), even or odd.

Further edit Feb. 22 : Interestingly, the case of odd lattices seems to allow the opposite behaviour:

p= 7  A=
            [ 7  0  0  0]
            [ 0  3  1 -1]
            [ 0  1  2 -1]
            [ 0 -1 -1  2]
fix= 0    move= 2   proportion= 0.000

p= 23  A=
            [23  0  0  0]
            [ 0  5 -1  0]
            [ 0 -1  3 -1]
            [ 0  0 -1  2]
fix= 0    move= 4   proportion= 0.000

p= 31  A=
            [31  0  0  0]
            [ 0  6 -1 -1]
            [ 0 -1  3  0]
            [ 0 -1  0  2]
fix= 0    move= 8   proportion= 0.000

p= 47  A=
            [47  0  0  0]
            [ 0  6 -2  1]
            [ 0 -2  5  0]
            [ 0  1  0  2]
fix= 0    move= 12   proportion= 0.000

p= 71  A=
            [71  0  0  0]
            [ 0  7  1  1]
            [ 0  1  6  1]
            [ 0  1  1  2]
fix= 0    move= 20   proportion= 0.000

p= 79  A=
            [79  0  0  0]
            [ 0  6  0  1]
            [ 0  0  5 -1]
            [ 0  1 -1  3]
fix= 0    move= 26   proportion= 0.000

p= 103  A=
            [103   0   0   0]
            [  0  18  -1  -1]
            [  0  -1   3   0]
            [  0  -1   0   2]
fix= 0    move= 50   proportion= 0.000

p= 127  A=
            [127   0   0   0]
            [  0  10  -2   1]
            [  0  -2   5  -1]
            [  0   1  -1   3]
fix= 0    move= 62   proportion= 0.000

p= 151  A=
            [151   0   0   0]
            [  0  11  -1   0]
            [  0  -1   5   1]
            [  0   0   1   3]
fix= 0    move= 78   proportion= 0.000

p= 167  A=
            [167   0   0   0]
            [  0  18  -2  -1]
            [  0  -2   5   0]
            [  0  -1   0   2]
fix= 0    move= 92   proportion= 0.000

p= 191  A=
            [191   0   0   0]
            [  0  14   0   1]
            [  0   0   5  -1]
            [  0   1  -1   3]
fix= 0    move= 114   proportion= 0.000

p= 199  A=
            [199   0   0   0]
            [  0   7   1   1]
            [  0   1   6   0]
            [  0   1   0   5]
fix= 0    move= 134   proportion= 0.000

p= 223  A=
            [223   0   0   0]
            [  0  11   3  -3]
            [  0   3   6  -2]
            [  0  -3  -2   5]
fix= 0    move= 182   proportion= 0.000

p= 239  A=
            [239   0   0   0]
            [  0  19   2   0]
            [  0   2   7  -1]
            [  0   0  -1   2]
fix= 0    move= 172   proportion= 0.000

Edit Feb. 21 : Let $p$ be a prime, congruent to $3$ mod $4$. Then there is such a genus on the $\mathbf Q$-quadratic space $[1,1,p,p]$. Here is the number of fixed (resp exchanged) lattices:

Original post :

Here is a final example : on the $\mathbf Q$-quadratic space $<1,2>\perp p.<1,2>$ (here, the example in the OP, developped in my first answer, appears):

p=  5  fix= 1    move= 0   proportion= 1.00
p=  7  fix= 1    move= 0   proportion= 1.00
p=  13  fix= 1    move= 0   proportion= 1.00
p=  23  fix= 6    move= 0   proportion= 1.00
p=  29  fix= 6    move= 0   proportion= 1.00
p=  31  fix= 6    move= 0   proportion= 1.00
p=  37  fix= 2    move= 2   proportion= 0.500
p=  47  fix= 15    move= 0   proportion= 1.00
p=  53  fix= 9    move= 2   proportion= 0.818
p=  61  fix= 9    move= 2   proportion= 0.818
p=  71  fix= 28    move= 0   proportion= 1.00
p=  79  fix= 20    move= 2   proportion= 0.909
p=  101  fix= 35    move= 2   proportion= 0.946
p=  103  fix= 25    move= 6   proportion= 0.807
p=  109  fix= 15    move= 12   proportion= 0.556
p=  127  fix= 30    move= 12   proportion= 0.714
p=  149  fix= 49    move= 12   proportion= 0.804
p=  151  fix= 49    move= 12   proportion= 0.804
p=  157  fix= 21    move= 30   proportion= 0.412
p=  167  fix= 88    move= 6   proportion= 0.937
p=  173  fix= 56    move= 20   proportion= 0.737
p=  181  fix= 40    move= 30   proportion= 0.571
p=  191  fix= 117    move= 6   proportion= 0.951
p=  197  fix= 45    move= 42   proportion= 0.518
p=  199  fix= 81    move= 20   proportion= 0.802
p=  223  fix= 70    move= 42   proportion= 0.625
p=  229  fix= 50    move= 56   proportion= 0.472
p=  239  fix= 165    move= 12   proportion= 0.933
p=  263  fix= 156    move= 30   proportion= 0.839
p=  269  fix= 132    move= 42   proportion= 0.759
p=  271  fix= 132    move= 42   proportion= 0.759
p=  277  fix= 36    move= 110   proportion= 0.247
p=  293  fix= 117    move= 72   proportion= 0.619
p=  311  fix= 266    move= 20   proportion= 0.930
p=  317  fix= 70    move= 132   proportion= 0.347
p=  349  fix= 105    move= 132   proportion= 0.443
p=  359  fix= 304    move= 42   proportion= 0.879
p=  367  fix= 144    move= 132   proportion= 0.521
p=  373  fix= 80    move= 182   proportion= 0.305
p=  383  fix= 289    move= 72   proportion= 0.801
p=  389  fix= 187    move= 132   proportion= 0.586
p=  397  fix= 51    move= 240   proportion= 0.175
p=  421  fix= 90    move= 240   proportion= 0.273
p=  431  fix= 399    move= 72   proportion= 0.847
p=  439  fix= 285    move= 132   proportion= 0.684
p=  461  fix= 300    move= 156   proportion= 0.658
p=  463  fix= 140    move= 272   proportion= 0.340
p=  479  fix= 525    move= 72   proportion= 0.880
p=  487  fix= 147    move= 306   proportion= 0.325