Question: What would the reconstructed signal look like if I instead used a Gaussian reconstruction filter?
Gaussian reconstruction filter will make the reconstructed signal smoother than the triangle(tent) filter. This is because Gaussian filter decays exponentially as opposed to triangle filter's linear decay.
As another possibility, would the reconstructed signal basically look like the original signal (perhaps without the fat upward initial tail at $T = 0$)? The initial curve already looked "fairly Gaussian" so I'd imagine that using a Gaussian filter would be a fairly close approximation.