Try minimizing the square of the difference.
alpha=.;t=.; (* remove any cached previous values *)
Quiet[
NMinimize[{
((1-4 NIntegrate[t^(1-1/alpha)(1-t)^(1/alpha),{t,0,1/2}]/
NIntegrate[t^(-1/alpha)(1-t)^(1/alpha),{t,0,1/2}])-0.2455)^2,1<alpha<5},
alpha]]
which returns
{4.52028*^-18,{alpha->2.93201}}
and
alpha=.;t=.;
ListPlot[Table[{alpha,((1-4 NIntegrate[t^(1-1/alpha)(1-t)^(1/alpha),{t,0,1/2}]/
NIntegrate[t^(-1/alpha)(1-t)^(1/alpha),{t,0,1/2}])-0.2455)^2},{alpha,1,5,1/8}]]
which will give you an idea how this behaves.
The reason for the Quiet
is that it complains about complex numbers for some of your powers. Usually I don't like hiding warning messages when doing things like this, but this can at least show you a result.
Remember there are always at least six different ways of doing anything in Mathematica.