When you hardcode the value, you will get “better estimates of the LTBI annual risk” only if the hard-coded value is actually exactly correct. If it’s not exactly correct you get biased values for LTBI that reflect this assumption about ATBI.
Choosing a range for ATBI that is consistent with the literature and broad enough to cover all values you are willing to consider will give you a wider spread of LTBI risk values which accurately reflects the uncertainty
If you run your model with ATBI as a parameter with some tight prior, and you want to make statements about the case where ATBI=0.1 you can do this by simply taking a sub-sample of your Stan output where ATBI is in the range say 0.09 to 0.11 and describing the LTBI for this sub-sample. Taking a largish number of iterations will help you make this information more accurate.
So, in some sense, the prior over ATBI includes the ATBI=0.1 as a sub-model and you can get that information out of it if you simply ask Stan for a large enough sample, while at the same time, you can compare this value to the range of values found when ATBI ranges over the full prior range. To make this work you might well want say iter=10000 or something like that so that there are several hundred samples in your sub-range.