Let $0<p<\frac{1}{2}$. I am looking for the limit:
$$\lim_{t \to \infty} \left(\frac{t}{\frac{t}{I_{2 p}^{-1}\left(\frac{t}{2},\frac{1}{2}\right)}-2 \sqrt{t} \sqrt{\frac{1}{I_{2 p}^{-1}\left(\frac{t}{2},\frac{1}{2}\right)}-1}+1}\right)^{t+1}$$ where $I^{-1}_{(.)}(.,.)$ is the Inverse Regularized Beta Function.
Numerically the convergence is extremely fast:
Due to the $t+1$ exponent, the function will converge quite slowly. If you try extremely larges values of $t,$ you'll see the apparent asymptotes are illusory.
Note that $\frac1{I^{-1}_{2p}(t/2, 1/2)}$ seems to converge to $1$ as $t \rightarrow \infty$ for $0 \leq p,\space p= 0.5$; that should help.
If that holds up, the limit will be $\frac1{e}$.