The way to dissuade the sampler from visiting values that are smaller than you want is to formulate a hard constraint if it’s a logical constraint (like probabilities falling between 0 and 1 or correlations between -1 and 1 or covariance matrices being positive definite) and as a soft constraint with a prior if it’s not.
Trying to hack terms into the density like this is almost always going to be a bad idea as it destroys the generative nature of whatever the underlying model is.