How to do a functional derivative with Exp and integration？

Define a functional derivative as:

[ScriptCapitalD][functional_,f_[y_]]:=Assuming[Element[y,Reals],Limit[(
(functional/.f:>Function[x,f[x]+[CurlyEpsilon] DiracDelta[x-y
]])-functional)/[CurlyEpsilon],[CurlyEpsilon]->0]]

And what I want to do is the functional derivative with exponential and integration like:

[ScriptCapitalD][[ScriptCapitalD][[ScriptCapitalD][[ScriptCapitalD][Exp[!(*SubsuperscriptBox[([Integral]), (-[Infinity]), (+[Infinity])]((-J[x]) Df[x - y] J[y]/2 [DifferentialD]x [DifferentialD]y))], J[x4]], J[x3]], J[x2]], J[x1]] /. {J[x] -> 0, J[y] -> 0}

Maybe mathematica can not do the derivative firstly and the do the integration, so I remove the Integration in the Exp:

[ScriptCapitalD][[ScriptCapitalD][[ScriptCapitalD][[ScriptCapitalD][Exp[-J[x] Df[x-y] J[y]/2],J[x4]],J[x3]],J[x2]],J[x1]]/.{J[x]->0,J[y]->0}

And the result is given by:

1/8 (4 Df[x-y]^2 DiracDelta[x-x3] DiracDelta[x-x4] DiracDelta[-x1+y] DiracDelta[
-x2+y]+Df[x-y]^2 DiracDelta[x-x2] DiracDelta[x-x4] DiracDelta[-x1+y] DiracDelta[
-x3+y]+4 Df[x-y]^2 DiracDelta[x-x1] DiracDelta[x-x4] DiracDelta[-x2+y
] DiracDelta[-x3+y]+4 Df[x-y]^2 DiracDelta[x-x2] DiracDelta[x-x3
] DiracDelta[-x1+y
] DiracDelta[-x4+y]+4 Df[x-y]^2 DiracDelta[x-x1] DiracDelta[x-x3] DiracDelta[-x2+y] DiracDelta[-x4+y]+4 Df[x-y]^2 DiracDelta[x-x1] DiracDelta[
x-x2] DiracDelta[
-x3+y] DiracDelta[-x4+y])

But how to do the integrations again? The orders of DiracDelta can not be distinguished.

Or are there any other solutions?

P.S. The basic question here is functional quantization of scalar fields, you can refer to P291 in Peskin’s textbook An Introduction to Quantum Field Theory.