While playing with this, I got:
Error, (in SumTools:-DefiniteSum:-ClosedForm) summand is singular in the interval of summation
I don't see how this can be singular, as Beta(k,1/2) is trivially bounded for k>=1, and sum(1/(2*k+1)^2) is absolutely convergent.
Maple is still able to evaluate numerically the inert sum (I get 0.3361376233). If I replace Beta with the expression with GAMMA, I get a result with MeijerG, that doesn't seem to help much for numerical evaluation, as it takes a long time and gives the wrong result -1.966769953*10^10 - (6.886827095*10^8)*I.
I have no reason to expect a closed form for this sum, but the error is surprising. And I don't know MeijerG well enough to explain the numerical error either. It fails with Maple 2020, 2021 and 2022.