Hmm, a couple of key things to consider here.
Interchangeable lenses - Most venues that prohibit "professional" cameras define professional as having an interchangeable lens, which the Fuji you suggested has. If you're not worried about taking photos as shows or getting your camera in then these cameras can run a good balance between sensor size (which is tied to both image quality and low light performance) and portability. The Sony A6X00 series (I have a 6600 for a travel camera) I also find a good performer. The A6400, which is essentially the same camera and sensor but without image stabilization and a couple of other features, could also be a good choice, however, I find IS is critical, especially if your goal is to take shots at a distance with low light. Keep in mind that all the cameras here are just camera bodies and you can spend any amount on lenses and the cost is generally proportional to the zoom range and the size of the aperture (the amount of light they let in) so your budget can get pretty large here pretty quickly. Also, even though these are relatively compact body sizes for APS-C size sensors, lenses with more zoom and larger apertures also tend to be quite large so this is something to consider based on what is most important to you. [Edited to correct zoom range of lens and add link]: I like my sigma 18-50 F2.8 with the A6600 as it give me 27-75 zoom range (35mm equivalent) with a large aperture, small size and weight and reasonable cost. They also make these for Canon and Fuji mounts I believe.
https://www.bhphotovideo.com/c/product/1667228-REG/sigma_18_50mm_f_2_8_dg_dn.html. They also have 3 nice 1.4 primes, however, they end up being bigger than the 18-50 so I tend to use the zoom to keep the size manageable.
Point and shoot cameras - These are allowed in pretty much anywhere and are "pocketable". The advantage of these over the above is the portability and the ability to get them in anywhere. However, this market is almost gone at this point due to the advancement in smartphone cameras. Typically, the advantage with a premium point and shoot would be low light performance and optical zoom, along with the
possibility of a view finder. My last purchase of one of these was a canon about 6 years ago which had good zoom, image stabilization and decent low light performance. It also had a viewfinder and an articulating back screen which was useful for taking pictures at shows as I could rotate the screen down so I could compose the shot while the camera was above my head in the crowd. I believe it was a Canon powershot G7X. It looks like they still make them however, the one I got was the last one with the viewfinder which I use for the vast majority of applications. The sony RX100 line is pretty impressive in this space and there have been like 8 version or so at this point. This
https://www.pcmag.com/picks/sony-rx100-buying-guide-which-high-end-compact-camera-is-right-for-you page gives a breakdown of the different versions. One of the key differences to look out for is the VA to the VI where they added a significant amount of zoom (good for shows) at the expense of the amount of light (bad for low light shows) so again, depending on your use case, you will likely have a preference.
Anyhow, my thoughts from 50,000'. I'm sure there are other models that you should be considering but prioritizing your use cases will be important to narrowing the field.
I hope that this is helpful!
edit: Beatkilla responded while I was typing this out. We didn't even discuss larger sensor cameras (DSLRs) as I assumed you were looking for something small that you could travel with easily and take to shows. DSLRs are great too, better image and low light at the trade off of portability and cost (lenses typically cost more) so again, use case dependent...