Since the reviewer only seems to be concerned about the two outcomes measured on the same subjects (and did not question the modeling procedure itself), I would simply use a sequential Bonferroni adjustment (a.k.a. Holm-Bonferroni method) to correct for it.
- Sort your $p$-values in ascending order
- Refer to them as $p_i$ (i.e. $p_1, p_2, p_3$, etc.)
- Than you adjust your $\alpha$-level and compare the $p$-values against that new $\alpha$-levels, i.e. you test whether $p_i \le \alpha / (1 + k - i)$, where $k$ is the number of statistical tests conducted, i.e. the number of $p$-values calculated. You can stop when $p_i \gt \alpha / (1 + k - i)$. Those $p_i$ that fall below the sequentially adjusted $\alpha$-levels are now your significant tests which are adjusted for multiplicity (after the Holm-Bonferroni method).
For example you conducted five tests ($\alpha = 0.05$) resulting in the following $p$-values:
$p_1 = 0.0024, p_2 = 0.0084, p_3 = 0.019, p_4 = 0.027, p_5 = 0.12$
The new $\alpha$-level you compare $p_1$ against is:
$0.05/(1+5-1) = 0.01$
Since $p_1 \le 0.01$ you can move on to $p_2$:
$0.05/(1+5-2) = 0.0125$
Since $p_2 \le 0.0125$ you can move on to $p_3$:
$0.05/(1+5-3) = 0.0167$
Since $p_3 \gt 0.0167$ you can stop.
In this case, from initially four significant $p$-values, you now only have two but those are adjusted for multiplicity (Note: Instead of adjusting the $\alpha$-levels, you can also adjust the $p$-values and compare against your chosen $\alpha$-level (e.g. $\alpha = 0.05$). Then all you need to do is $(1 + k - i)*p_i$ instead).
See also:
Abdi, H. (2010). Holm’s sequential Bonferroni procedure. Encyclopedia of research design, 1.
Peres-Neto, P. R. (1999). How many statistical tests are too many? The problem of conducting multiple ecological inferences revisited. Marine Ecology Progress Series, 176, 303-306.
Alternatively, you could also argue that you don't want to adjust for multiplicity because of reason such as being concerned with making type II errors.
See here:
Feise, R. J. (2002). Do multiple outcome measures require p-value adjustment?. BMC Medical Research Methodology, 2(1), 1.
Or maybe this one: Gelman, A., Hill, J., & Yajima, M. (2012). Why we (usually) don't have to worry about multiple comparisons. Journal of Research on Educational Effectiveness, 5(2), 189-211.
k=2
option instepAIC
? $\endgroup$