The subtility in the OP's question is in the use of dimensionfull and dimensionless quantities. This is easier to understand if one does not work at the upper critical dimension ($d=4$ in most standard field theories), where the interaction has dimension (even using dimensionfull quantities). I will first comment on the case where the UV cut-off $\Lambda$ is finite, and then comment on the limit $\Lambda\to \infty$.
To give an example, let us look at a massless $\phi^4$ theory in $d=5$, which would correspond to a trivial theory in a renormalization group sense, the infrared fixed point being the gaussian fixed point. This, however, does not mean that the renormalized parameter of the theory are all zero !
The most interesting quantities are low energy quantities, which correspond to the "renormalized parameters" of the OP, like the (renormalized) mass $m_R$ (here equal to zero) and the (renormalized) interaction $g_R$. Physically, $g_R$ corresponds to a scattering amplitude at zero 5-momentum $g_R$, and it is a priori finite (that is, unless the bare theory is non-interacting), as one could compute in a perturbative expansion in $g_B$, the bare value of the interaction. (Note that $g_R$ will depend on $\Lambda$, but that is not a problem, since $\Lambda$ is assumed finite.)
Now, what do we expect in a renormalization group point of view? First, we need to use dimensionless quantities, and introducing $\tilde g= \mu g$, we find that after a short transient regime (corresponding in the dimensionfull units to the transition from $g(\mu=\Lambda)=g_B$ to $g(\mu\ll\Lambda)=g_R$), the renormalized dimensionless interaction flows to zero (i.e. to the gaussian fixed point) $\tilde g(\mu)\propto \mu$, while the dimensionfull interaction is constant.
The OP was interested in the case of a relevant variable, which can be included easily in the example by assuming that the theory is massive, but with a very small (dimensionfull, renormalized) mass $m_R\ll \Lambda$. To study the RG flow, we need a dimensionless mass $\tilde m = m/\mu$, and we see that in the IR, $\tilde m$ is a relevant perturbation, that diverges as $\mu\to0$ (although, of course, the "real" mass stays finite and equal to $m_R$).
Let me now comment briefly on the issue of the limit $\Lambda\to\infty$ and the "infinite" value of the bare parameter, which is in my view an artificial problem introduced by an old-school approach to field theory (and unfortunately still promoted in textbooks and pop-science). In the present case (as well as in $d=4$), we cannot take the limit $\Lambda\to\infty$ while keeping $g_R$ finite, as there is no UV fixed point to control the flow at high-energy, and the only consistent theory is the free theory $g_B=0$. I do not really want to go further into this subtle problem (and how one could work this out $d=3$), but a related discussion is given here : Why do we expect our theories to be independent of cutoffs?