numerics – How to approximate the harmonic sum $sum_{n=1}^inftyfrac{{4nchoose 2n}overline{H}_{2n}}{n 2^{4n}} ?$

How to approximate $$sum_{n=1}^inftyfrac{{4nchoose 2n}overline{H}_{2n}}{n 2^{4n}} ?$$
Where $overline{H}_n=sum_{k=1}^n frac{(-1)^{k-1}}{k}$ is the skew harmonic number.
The mathematica command for $overline{H}_{2n}$ is $log(2)-text{LerchPhi}(-1,1,2n+1)$.

I tried Michael E2′ command:

 major = Normal@Series((Log(2) - LerchPhi(-1, 1, 2 n + 1)) Binomial(4 n, 
   2 n)/(n 2^(4 n)), {n, Infinity, 12});
 majorsum = Sum(major, {n, Infinity});
 majorsum + 
 NSum((Log(2) - LerchPhi(-1, 1, 2 n + 1)) Binomial(4 n, 
  2 n)/(n 2^(4 n)) - major, {n, 1, Infinity}, NSumTerms -> 20, 
 WorkingPrecision -> 20, Method -> "WynnEpsilon")

but it gave a result in terms of $n$ which something unusual to see.

Is there any other command or maybe can do little changes in Michael E2’s solution?

Thank you,