Is there a #$P$-complete counting problem such that every (valid) instance of its decision version is a Yes-instance?
An example of a such problem in $\mathrm{DTIME}(n)$ is DNF satisfiability [1]. Its counting version is #P-complete [2], but its decision version can be solved by simply checking if there is a conjunction that is satisfiable.
For $\mathrm{DTIME}(1)$ I think the answer is that there are no such problems because we can iterate over all computation paths in $O(1)$ so counting is easy.
Note that in general, disallowing instances with answer 0 does not change the #P-completeness of any problem whose decision version is in P because we can check this case in polynomial time.
[1] https://en.m.wikipedia.org/wiki/Disjunctive_normal_form
[2] https://en.m.wikipedia.org/wiki/%E2%99%AFP-complete