Thanks for your input everyone. I decided to go for GI Effects Comprehensive Profile by Genova Diagnostics. My doctor (a private doctor specialised in gut health) took a look at the most comprehensive GI 360 test and she claimed there wasn't that much of a difference. Also, it was way more convenient for me to do the Genova one as I was able to drop it off the stool samples at a local lab. Might do the GI 360 one later if I still feel like the gut angle is promising. Will have the results in 3 weeks. I'll post them here.
On the debate about whether microbiome problems are worth pursuing. Based on what I've read so far, and my symptom picture, it definitely seems plausible that an unhealthy gut microbiome could be the root cause. The trouble though, which I raised with my doctor, seems to be that we know so little about the gut microbiome at this point. AFAIK there are lots of bacteria that haven't been mapped out yet, and lots we don't know the function of. My doctor's response was that sure, there's a lot we don't know, but there are certain specific bacteria or families of bacteria that are either proven to be pathogenic or proven to be beneficial. So the presence or absence of those might be helpful to know about, and if things are clearly in a bad state, you can consider a faecal transplant or other treatments.