I am conducting a multiple linear regression on data from a cross-sectional study, and I suspect that there is an interaction between my dependent variable (a disease risk marker) and one independent variable (an exposure). Biologically, this would make sense: the compound I am interested in (a type of flame retardant) can affect blood cholesterol concentrations - and the effect seems to be stronger in those with already elevated plasma lipids.
My initial analyses used an approach (y: endpoint - here cholesterol, $x_1$: exposure - flame retardant):
$$ y = a + \beta_1 x_1 + ... $$
But I know now that there is some relationship $ \beta_1 \sim y $, and when I stratified by quantiles of $y$, $\beta_1$ changes from about -1 to +1 (there are no differences in distribution of $x_1$ between quantiles). So actually the model should include an interaction between $x_1$ and $y$:
$$ y = a + \beta_1 x_1 y + ... $$
What is the most appropriate way to address this?