There have been refined notions of what it means for a protocol to have $\lambda$-bits of security. The best-known one is probably the Micciancio-Walter On the Bit Security of Cryptographic Primitives.

The notion of bit security there is different whether one is working with a

decision game, where it is $\min_A\log_2 \frac{T(A)}{(\mathsf{adv}^A)^2}$, i.e. scales *quadratically* with the adversarial advantage, or

search game, where it is $\min_A \log_2 \frac{T(A)}{\mathsf{adv}^A}$, i.e. scales *linearly* with the adversarial advantage.

The $\min_A$ is minimizing over all adversaries attacking a primitive, and $T(A)$ is the running time of $A$.
Note that advantage for search/decision games are defined differently (for search games, it is roughly the success probability, for decision games, it is $2\delta-1$, where $\delta$ is the success probability).

Anyway, in this framework you could try to prove something like

If $H(\cdot)$ is a hash function that is $\kappa$-bit pre-image resistant, then $H^{\circ k}(\cdot)$, the $k$-fold composition of $H$, is $(\log_2(k) + \kappa)$-bit pre-image resistant.

The proof roughly go as follows

Start with $\min_A \log_2 \frac{T(A)}{\mathsf{adv}_H^A}\geq \kappa$, where $\mathsf{adv}^A_H$ is the advantage of $A$ in a game defining pre-image resistance.

Establish some sort of advantage bound $\mathsf{adv}_{H^{\circ k}}^A \leq \mathsf{adv}_H^A$, keeping track of the running time of the new adversary $B$ one (implicitly) constructs as part of this. For simplicity, assume that $\mathsf{adv}_{H^{\circ k}}^B \leq \mathsf{adv}_H^A$, and $T(B) = kT(A)$ --- it is not clear to me this is true, but I think this assumption is implicit in your question.

Then, we have that $\min_B \frac{T(B)}{\mathsf{adv}_{H^{\circ k}}^B} \geq \min_A\frac{kT(A)}{\mathsf{adv}_{H}^A} = \log_2 k + \kappa$.

As #2 above points out, this really reduces to showing a (traditional) advantage bound of the form.

For any adversary $B$ against $SHA3^{\circ k}$, there is an adversary $A$ against $SHA3$ with

- $T(B) = kT(A)$, and
- $\mathsf{adv}^B_{SHA3^{\circ k}} \leq \mathsf{adv}^A_{SHA3}$.

This is to say that the MW framework allows one to discuss how much the higher running time of $B$ impacts security (which is your question), but one still needs to prove an advantage bound though.