To get radiation fields, you need Maxwell's equations and boundary conditions. When you have those, what is the use of GA?

1) Start with the Coulomb potential A(r) = qγ₀/(4πr)

2) make it covariant by replacing r -> r -> X⋅γ₀ where X = r(γ₀ + γᵣ). This is a trivial transformation in the static frame as one can easily show X⋅γ₀ == r, but it's important as it encodes the causal information you astutely pointed out is vital to obtaining radiation solutions.

This gives A(r) = qγ₀/(4π X⋅γ₀)

3) Now, one can 'rotate' this to a frame with (not necessarily constant) velocity v via A'(r) = R A(R̃ x R) R̃ ie. evaluate the rotated A at inverse rotate spacetime position. Then

A'(r) = q v/(4π X⋅v)

This is the Liénard–Wiechert potential (this can also be obtained by solving ∇²A = J using the retarded Green's function G(r, t) = δ(|r|-t)/(4π|r|).

4) Now once can get the radiation fields by taking the derivative of this and doing some algebra:

F = ∇A = (q/4π) (X ∧ v + ½ X(v̇ ∧ v)X)/(X⋅v)³

The term proportional to X ∧ v is just the Coulomb field, but the term with the X(v̇ ∧ v)X is the radiation field which one can show propagates to infinity.

This is unfortunately much less elegant and straightforward than I let on initially so I apologize.

I still think this has advantages over standard treatments but your milage may vary.