I can give my best guess, FWIW. I think drinking a whole bottle and having no reaction means you don't have a major toxicity problem like me--I've avoided kombucha for this reason. But I also believe there's a wide spectrum between good, healthy liver function and people like me, and most everybody should think about liver function and how their particular bodies detoxify (lurking in Allergies right now is helpful, I know you've been over there, Meta, but it's working to help peoples' food intolerances, yay! and I've seen ups/downs in my son's intolerances based on his liver being more/less stressed too). People have strengths and weaknesses, and if you can figure out clues to what your particular body needs, you'll be WAY ahead of the game in terms of feeling good and staying healthy long-term, and for you I bet it'll help the rate at which your adrenals heal.
I did this while nursing on the advice of my HCP, and a) since vitamin C is so important for the adrenals, it's supportive that way, and b) it lowers circulating toxins, and oftentimes toxins seem to settle more easily in the adrenals and thyroid than other places (don't know why) so it would be supportive that way too...
http://www.perque.com/pdfs/Pt_Ascorbate_Slush_FIN.pdf
Fun? Well, not really (read the whole doc, you'll understand why) but IMO (and in my HCP's opinion, and she was pretty conservative on what nursing moms should do) it's safe while nursing, and I know I've felt better doing these. But, I think that's partly because I'm more messed up--you may not feel better right away, but I think they are supportive and help with detoxification by reducing the load on the liver.
So it's possible that your liver is stressed, but it's likely that you haven't come across a major source that you can't detoxify well. So you're not me.

In general, we get detox reactions when there's a big disconnect between what's mobilized and what we have the bandwidth to excrete. For me, I have a tendency to not excrete heavy metals well and have accumulated quite a high amount, and so smaller amounts of things that mobilize toxins have the potential to mobilize a whole lot in me (until I get my toxic load down), and my excretion bandwidth is/was fairly impaired. But there's a window in which you've mobilized stuff, and you can't excrete it all yet, but your body can handle it (more circulates for a while, some may re-settle) but you feel okay--you don't have headaches, body aches, other weird stuff. So it doesn't imply that the kombucha didn't mobilize anything in you (and so at least some of what was mobilized was excreted, so you've come out ahead), or that your liver function is peachy keen, but the scale of the potential problem is pretty manageable.