Uncertainty of individual patient pharmacokinetic parameters

In daily practice we are often asked to estimate the ellimenation rate constant of a drug in a patient in order to choose a suitable dosing regimen or calculate 24- hour area under the concentration time curve. Usually , one either uses so called " Bayesian MAP" estimation or we get two blood samples after a dose , at steady state, and calculate the ellimenation rate constant assuming mono-exponential decay.
My question concerns the later method. In clinical practice the elimination rate constant is calculated as:
K = log (C2/C1)/(t2-t1)
C= observed concentration
now if you have only two blood samples of drug concentration one can get a point estimate of K but since there are zero degrees of freedom the uncertainty cannot be calculated ( from a linear regression model) and reported or used for decision making.
Is there a way to quantify the uncertainty in this case? taking 3 blood samples after a dose from a patient is too painful to the patient and time consuming for healthcare professionals.

Thanks for you suggestion. the concept you suggested is also used for the reporting of the uncertainty of GFR ( estimated with the MDRD formula for example).

I thought about a solution and would like to know what others think.
set a normal prior on K , normal(mu,sigma) , mu and sigma based on prior knowledge of values of K in patients with similar characteristics, with an upper bound at zero ( the concentration is decreasing) and use Bayesian linear regression , with greater number of iterations.
I tried it with “brms” and got reasonable results. is this a valid method to estimate the uncertainty about this paramater?