A mixed model is not relevant here. A simple linear regression with one variable will achieve exactly the same results. Coding it as -1 and 1 has no difference to coding it as 0 and 1. You just stuff the rest into the intercept.
You would also want to be predicting 0.45 and 0.55 not 1 and 0 because we solve for squared error.
You would also want to be predicting 0.45 and 0.55 not 1 and 0 because we solve for squared error.