Controlling for phylogenetic signal - what is statistically appropriate?
I have a tentative answer to my question. 

In the publication for the R package phylosignal (Keck, 2016), they state:


  To test the presence of phylogenetic signal, the null hypothesis is that trait values are randomly distributed in the phylogeny. Another null hypothesis might be that trait values follow a Brownian motion model but it is less often used and implemented.


So it would seem there are indeed two null hypotheses, however it seems disingenuous to not test for both, especially since it is relatively simple to do. And of the two, should you for some reason be limited to one, testing for $\lambda = 0$ (or the log-likelihood of a general linear model) should be your first choice. 

Another issue was with corPagel from the package ape. It requires an initial value to estimate Pagel's $\lambda$ (Unless you fix the value, of course). Convergence is not guaranteed, and sometimes requires some fine-tuning of the initial value. This set off a bit of a red-flag, so I collected the median lambda from 5000 subsets of my data (I am using a different family/phylogeny/dataset, where the $\hat{\lambda}$ model is preferred over a Brownian, but not $\lambda = 0$).  

    

I think this is stronger argumentation for a phylogenetic signal, albeit a weak signal.

#output of comparing lambda=0, lambda=1, and median lambda

                 Model df      AIC      BIC    logLik
pglsModel_0          1  3 636.9876 643.6899 -315.4938
pglsModel_1          2  3 681.9784 688.6807 -337.9892
pglsModel_MEDIAN     3  3 636.4812 643.1836 -315.2406

#output of comparing median lambda to the estimated lambda from the full dataset

                 Model df      AIC      BIC    logLik   Test   L.Ratio p-value
pglsModel_MEDIAN     1  3 636.4812 643.1836 -315.2406                         
pglsModel_FULL       2  4 637.7024 646.6388 -314.8512 1 vs 2 0.7788265  0.3775


By the evidence of the distribution of lambdas, and the median model's AIC, BIC, and log-likelihood, we can argue in favor of the median model. Residuals also look normal and random. Confidence intervals should also be obtained. But, in reality, here, the difference here between a weak phylogenetic signal and none at all are vanishingly small.