Computing the continuous time survival rate
I think the step


  "...where $P(a, t) = exp(-d(a,t)\Delta)$ is the discrete time analogue of $d(a,t)$..."


is the problem.

In continuous time I guess we have

$$\dot m(a,t) = -d(a,t)m(a,t) \implies m(a,t) = m_0\exp \{-d(a,t)t\}$$

We then have, discretizing,

$$\frac {m(a,t+\Delta) - m(a,t)}{m(a,t)} = \frac {\exp \{-d(a,t)(t+\Delta)\}-\exp \{-d(a,t)t\}}{\exp \{-d(a,t)t\}}$$

$$=\exp \{-d(a,t)\Delta\}-1 \approx -d(a,t)$$

$$\implies d(a,t) \approx 1-\exp \{-d(a,t)\Delta\}$$

So it should be $P(a, t) = 1-\exp(-d(a,t)\Delta)$. Now $a$ changes exactly in the same way as $t$ does (at $t+\Delta$ those of age $a$ at $t$ will be of age $a+\Delta$). So $m$ changes in both arguments together.

$$m(a+\Delta, t+\Delta) = (1-P(a, t))m(a, t) = \exp(-d(a,t)\Delta m(a, t)$$

$$\approx [1-d(a,t)\Delta]m(a, t)$$

I understand the left hand side as "the people that were of age $a$ at time $t$, and are still alive at time $t+\Delta$, where they have become of age $a+\Delta$. Is this what you are after?