Does the linear probability model require the regressand to be zero/one-valued? Typically, the dependent variable in a linear probability model (LPM) is a 0/1-valued binary variable. What if the dependent variable $y_i$ is still binary but take on general values $a$ and $b$ rather than 0 and 1? Technically, the resulting predictor still retains its nice properties, e.g., it is the minimum mean squared error (MMSE) linear predictor. Again, we may also transform $y_i$ into a 0/1 variable; but sometimes we want to keep the original values, say, for interpretation.

My question is: Can we still call the estimator the LPM estimator? If not, what should we call it?
The "LPM" label refers to the structure of the equation, not to the estimator. LPM models can be estimated not only by least-squares methods but also by maximum-likelihood for example.  

As regards the nature  of the dependent variable, we are talking about an affine transformation here.  Let a model be with a binary dependent variable and a single regressor for simplicity,

$$y_i = \beta_0 +\beta_1x_i +e_i$$

with $y_i \in \{0,1\}$. We are then told that the original variable was 

$$z_i = a+(b-a)y_i$$

so $y_i=0\implies z_i =a$ and  $y_i=1 \implies z_i =b$.

Then the model for $z_i$ is

$$z_i = a+(b-a)[\beta_0 +\beta_1x_i +e_i]$$

or

$$z_i = \gamma_0 +\gamma_1x_i + u_i$$

$$\gamma_0 = a+(b-a)\beta_0,\;\;\; \gamma_1 = (b-a)\beta_1,\;\;\; u_i = (b-a)e_i$$

The structure of the model remains the same, an equation linear in parameters, so it is an LPM model still. 

Since $(a,b)$ are presumably known values, after estimating the $z$-model, and obtaining estimates for the gammas, we can recover the estimates for the betas also, if they are of interest.