you can only constrain the parameters or functions of parameters you are trying to optimize, you cannot really constrain something that is observed.

Lets assume $Q$ is one of the parameters and you can go on with this optimization. You already constrain your function such that $1^Tx - \sqrt{x^TQx} \leq 0$ since the distrb. you end up will satisfy the condition for all values, you will have a empirical prob. of 1. But this will be wrong since you are already forcing it to have prob. of 1.

Here I assuming you obtain the parameters $\Sigma$ and $\mu$ through MLE then use the CDF of estimated distrb. to find the probabilities.

You need to find the distrb. of $1^Tx - \sqrt{x^TQx}$ directly and that is tricky, even in the case you obtain the distrb. there is a good chance there is no close form solution for CDF.

One way I can think of, calculate $1^Tx - \sqrt{x^TQx}$ with the data you have, fit a kernel density estimator to values you obtain, then calculate the prob. through CDF of kernel density estimator.

Note that kernel density estimator has its own parameters to optimize.