Addendum

One of my colleagues pointed out that our WeierstrassHalfPeriod functions come equipped with known derivatives, which are closed under repeated differentiation. So there is a third or fourth way to obtain the general form of the period-annihilator coefficients:
In[]:=
testW3=D[WeierstrassHalfPeriodW3[{G2[x],G3[x]}],{x,#}]&/@{0,1,2};​​testW1=D[WeierstrassHalfPeriodW1[{G2[x],G3[x]}],{x,#}]&/@{0,1,2};
In[]:=
annW1=NullSpace[Coefficient[testW1,#]&/@{​​WeierstrassHalfPeriodW1[{G2[x],G3[x]}],​​WeierstrassEta1[{G2[x],G3[x]}]}][[1]];
In[]:=
annW3=NullSpace[Coefficient[testW3,#]&/@{​​WeierstrassHalfPeriodW3[{G2[x],G3[x]}],​​WeierstrassEta3[{G2[x],G3[x]}]}][[1]];
Apply[PolynomialGCD,annW1]
In[]:=
Divide[Together@annW1,Apply[PolynomialGCD,Together@annW1]]/.diffRules
Out[]=
-21G2[0]
3
G2[1]
G3[0]+18
2
G2[0]
2
G2[1]
G3[1]+8
3
G2[0]
G2[2]G3[1]-216G2[2]
2
G3[0]
G3[1]+108G2[1]G3[0]
2
G3[1]
-120G2[0]
3
G3[1]
-8
3
G2[0]
G2[1]G3[2]+216G2[1]
2
G3[0]
G3[2],(16(
3
G2[0]
-27
2
G3[0]
)(-3G2[1]G3[0]+2G2[0]G3[1])(-9
2
G2[0]
2
G2[1]
G3[0]+3
3
G2[0]
G2[2]G3[0]-81G2[2]
3
G3[0]
+7
3
G2[0]
G2[1]G3[1]+135G2[1]
2
G3[0]
G3[1]-108G2[0]G3[0]
2
G3[1]
-2
4
G2[0]
G3[2]+54G2[0]
2
G3[0]
G3[2]))((-
3
G2[0]
+27
2
G3[0]
)(3G2[1]G3[0]-2G2[0]G3[1])),16(
3
G2[0]
-27
2
G3[0]
)(-3G2[1]G3[0]+2G2[0]G3[1])
In[]:=
Divide[Together@annW3,Apply[PolynomialGCD,Together@annW3]]/.diffRules
Out[]=
-21G2[0]
3
G2[1]
G3[0]+18
2
G2[0]
2
G2[1]
G3[1]+8
3
G2[0]
G2[2]G3[1]-216G2[2]
2
G3[0]
G3[1]+108G2[1]G3[0]
2
G3[1]
-120G2[0]
3
G3[1]
-8
3
G2[0]
G2[1]G3[2]+216G2[1]
2
G3[0]
G3[2],(16(
3
G2[0]
-27
2
G3[0]
)(-3G2[1]G3[0]+2G2[0]G3[1])(-9
2
G2[0]
2
G2[1]
G3[0]+3
3
G2[0]
G2[2]G3[0]-81G2[2]
3
G3[0]
+7
3
G2[0]
G2[1]G3[1]+135G2[1]
2
G3[0]
G3[1]-108G2[0]G3[0]
2
G3[1]
-2
4
G2[0]
G3[2]+54G2[0]
2
G3[0]
G3[2]))((-
3
G2[0]
+27
2
G3[0]
)(3G2[1]G3[0]-2G2[0]G3[1])),16(
3
G2[0]
-27
2
G3[0]
)(-3G2[1]G3[0]+2G2[0]G3[1])