That appears to be where you have received all of your education on nuclear material.

You are just plain wrong. Let's say we've got a sample of that tritium and it reads 3 counts per second (CPS) on a lead-encased geiger counter (i.e. not exposed to background level). If I then take a non-lead encased counter outside, it now reads 10 CPS from normal atmospheric/environmental radiation. If I then take the tritium outside of the lead and measure it with the unshielded meter I will likely still get 10 CPS, i.e. no increase in background. The tritium in this case has less than the background radioactivity and therefore does not cause an increase, which means your dose does not increase.

Edit- this is an example, the CPS are made up to illustrate the point, lest you think I am stating exactly what will happen.

That kinda shows that you don’t understand how nuclear stuff works, quite frankly.