I believe the cause is due to "scoping", a programming jargon. So in this case, I don't think it is a solution for you by saving parameter estimates as table variables using the exact same names.
By scoping, the dynamic plot that you refer to has an "evaluation environment", which has two variables with the same name "Max" and "k". To see it, right click the plot, select "Customize", then select "Current Model Fit". You will see a formula, which uses "Max" and "k". Every time, when you click the sliders to change values of "Max" and "k", the program updates those values and re-evaluate the function. However, there is another set of "Max" and "k", which is in data table's "evaluation environment", which are not updated. Unfortunately, due to evaluation precedence, the formula sees them first and use them, even the set got updated is somewhere close-by. That explains why the plot does not change.
The only thing I can think of is to use either a prefix or postfix for the table variable names, or something that is easy for you to recognize, and variable names won't collide.