How to calculate step and impulse response at multiple sampling rates
This is not the full answer you want, but the problem is that you can't permute sampling and convolution. Let's say that the $D(\cdot)$ operator takes a continuous signal and returns its sampled version. What you're doing is

$$y'(n) = D\left(e^{-at}u(t)\right) * D\left(x(t)\right)$$

What you want to get is

$$D\left(y(t)\right) = D\left(e^{-at}u(t) * x(t)\right) \neq y'(n).$$

It is in general impossible to do what you're trying to do, but I'm sure that good approximations exist. I'm not an expert on what they are or how to use them.