How to increase price gradually?
The formula $t = \frac{e^2}{T^2 + e}$ does not scale linearly:

$\frac{(ke)^2}{(kT)^2 + ke} \ne c \frac{e^2}{T^2 + e}$ for any constants $k$ and $c$. (Linear scaling is OK as this is just a denomination of money. Nonlinear scale is a kind of nonsense as it introduces the need to add arbitrary coefficients, without a hint how to choose particular numerical values.)

So I choose the formula $t = \frac{e^2}{T + e}$.