Solving $\frac{M e^{-M}}{1-e^{-1}} - \epsilon = 0$ for $M \in \mathbb{R}$

I am trying to solve the following nonlinear equation analytically:$$\frac{M e^{-M}}{1-e^{-1}} - \epsilon = 0 \, ,$$where $ M \in \mathbb{R} $ and $ 0 < \epsilon \ll 1 $.

A solution can be expressed using the LambertW / ProductLog function as:$$M = -W\left( \epsilon (e^{-1}-1) \right) \, .$$

For $\epsilon = 10^{-6} $, one finds approximately $ M \approx 6.32 \times 10^{-7} $.

However, I am more interested in the solution where $ M \approx 17.11 $ using these numerical values. I am wondering whether an analytical formula can likewise be obtained for this latter case.