I am just trying to be helpful here, I am a doc, while no endocrinologist this stuff and particularly neuropsychiatric stuff is my bread and butter (cause I am insane and work out
). I say that just so I dont have to be posting 20 pubmed links as I am lazy and currently doing a teaching rotation 
It can elevate cortisol quite acutely, it's well documented. So much so that it can cause impaired memory and learning. Which is likely related to cortisols effect on the hippocampus.
https://pmc.ncbi.nlm.nih.gov/articles/PMC8957269/
Weight lifting is a hormetic stressor- technically damaging (so are fruits and veggies actually) but the response of the body is beneficial, only if and thats a big if, the allostatic load of the system can get back to allostasis. Heavy weights increase neural drive and recruitment and a higher HPA response because of the intensity.
His system with type 2 has systemic inflammation- its the likely cause of type 2 and most diseases- basically resistance causes oxidative stress and myokine/cytokine responses. The liver struggles to handle the load as well and that leads to all kinds of pro-inflammatory mediators being released into the system. Over time it worsens and beta cells begin to fail from oxidation amongst other things and you get type 2.
https://pubmed.ncbi.nlm.nih.gov/22252015/
you generally see higher crp, tnfa and nuclear factor kappa beta activation in type 2.
You are correct on the supplement, I was just fucking about.
What is likely going on here is he is lifting heavy, causing stress to the system which is already in overdrive, cortisol is released to dampen things, he has insulin resistance which cortisol is exacerbating leading to more insulin release then needed hence the fatigue and feeling shitty. There is likely a counter release of glucagon which does some odd shit to the body as well.
Oh, so you wanna get into a pubmed posting war, huh?

I can totally agree with the acute spike of cortisol, but again so many make cortisol out to be worse than agent smith. Yes, the heavy lifting can elevate it because the HPA axis thinks we’re trying to not get Clegged. We can agree that in a healthy system, that’s adaptive: cortisol mobilizes glucose and fats, then insulin cleans up the mess.
For Josh's type 2, the “cleanup crew” is on strike. What is interesting (and confusing to me) is that he is using metformin and berberine and shouldn't be getting this kind of spiking (safe assumption). I will admit that I am not highly knowledgable about the drug side of things. Could it be that with the low carb and then meds, his pancreas is still trying to pump out insulin, while glucagon is acting like DJ181818181??
HArd to argue that resistance training is not still the best way to fix the mess—it as it will improve insulin sensitivity and "lowers" inflammation—but in the short term the hormonal rollercoaster is a bit rougher when the system’s already overloaded.