Write your $\Lambda$ as
$$\Lambda(t)=\frac{1}{2}\sum_{-\infty}^\infty f(n),$$
where
$$f(x)=\pi(x+1/2)\sin\pi(x+1/2)\exp\left(-\pi^2(x+1/2)^2t/2\right)=y\sin y\,e^{-ty^2/2},$$
where $y=\pi(x+1/2)$.
Then use <a href="https://en.wikipedia.org/wiki/Poisson_summation_formula">Poisson's summation formula</a>
$$\sum_{-\infty}^\infty f(n)=\sum_{-\infty}^\infty \hat{f}(2\pi n),$$
where
$$\hat{f}(s)=\int_{-\infty}^\infty f(x)e^{-isx}dx$$
is the Fourier transform. This Fourier transform can be explicitly computed:
$$\hat{f}(s)=\frac{1}{\sqrt{2\pi}}t^{-3/2}e^{is/2}\left((s/\pi+1)e^{-(s/\pi+1)^2/(2t)}-(s/\pi-1)e^{-(s/\pi-1)^2/(2t)}\right).$$
The trick is that in the Fourier transform your parameter $t$ will stand in the denominator of the exponent, so the series $\sum\hat{f}(n)$ will be asymptotic to its zero-th term, when $t\to 0$,
$$\sum_{-\infty}^\infty\hat{f}(2\pi n)\sim \hat{f}(0)=\sqrt{\frac{2}{\pi t^3}}e^{-1/(2t)},$$
as you wrote. 

(In the computation of Fourier transform, I started with the $e^{-y^2/(2t)}$, and then used the transformation rules
of Fourier transform: multiplied my function on $y$, then on $\sin y$, and then scaled by $\pi$ and added $1/2$).