constrained dynamics - Significance of the the Lagrange multipliers in statistical mechanics

In classic thermodynamics one can derive the Maxwell Boltzmann statistics by solving a Lagrange multipliers equation. In this process a new parameter $\beta$ is introduced to take account of the total energy constraint,

$$L = \beta(E-\sum_i p_i \epsilon_i)+...$$

and it is later connected to $\frac{1}{k_B T}$ by inserting our knowledge about perfect gases. On a side note, about every book around seems to skip the passage with a frustrating "it can be shown that".

My problem is that, given similar Lagrange multipliers equation, I'm not sure whether one can in general relate $E$ and $\beta$: it is evident that $\beta$ has no physical significance by itself, and it's always bounded to some parameter in the constraints that it is multiplying.

For example, say that I have an array of cells that can be either occupied by 0, 1 or 2 dots. I have some constraints, one of which is the density of the dots $\rho$. The Lagrange multipliers equation has the form

$$L = \gamma(\rho-p_1-2p_2)+...$$

where $p_i$ is the probability of having a cell with $i$ dots. Is there some way to eliminate the $\gamma$ parameter in the final explicit formula for $p_1$, and use $\rho$ instead?