What properties does a co-RP problem need to be in P?
Considering from my comment that there exists an upper bound to the runtime of $A$: $p(n)$ then there are only $2^{p(n)}$ possible ways a coin-flip of a randomized algorithm can manipulate the outcome of the result.

If the error propability of $A$ I described as $f$ is a fraction with part $\frac{g(n)}{2^{p(n)}}$ then creating a deterministic program that runs $g(n)$ many times plus one different paths of a possible computation of $A$ it is guaranteed that at least one of these paths would have identified the false positive as false.

Therefore we can skip the journey through $RP$ into $ZPP$ and can argue that if $g$ is a polynomial function we need to run $g(n) + 1$ times the polynomial Algorithm $A$ resulting in a polynomial runtime of $(g(n)+1) \cdot p(n)$ that deterministically solves $x\in L$ or $x \not\in L$ with an error propability of $0$, therefore placing $L$ in $P$.

This is but one case where, given a randomized algorithm that places a language $L$ clearly in $co-RP$ places $L$ also in $P$.