How often do you make 16x24" or larger prints?
How often do you need to use ISO 12800 or higher?
If your answer to those questions is "never", a full frame camera would quite likely be of little or no benefit what-so-ever. You would probably be the only person to ever see any difference, while editing your RAW files at ridiculously large magnifications on your computer. A 24MP image at "100%" on most modern monitors is approx. equivalent to a 36x60" print... viewed from only 18 to 20" away. Most end uses of images are far smaller and/or viewed from a more realistic distance, so other people looking at your finished products are unlikely to see much, if any difference between a DX camera and an FX camera. In fact, it's arguable that the vast majority of FX/FF camera users actually get no benefit from their larger, heavier, more expensive camera and lens kits. Slightly slower flash/strobe syncs are common with FX/FF too (Portable flash sync of D600's is 1/200. It's 1/250 with D500.)
FX/FF is ideal for some things... landscape and architectural photography are a couple examples.
But for a lot of other things, DX/crop sensor cameras can be a better choice.
In fact, the best use of your money might be to put it toward more or better lenses, added lighting gear, backdrops, posing stands and props, etc.... for use with your current camera. A higher performance DX camera (D7200 or D500) might be a good upgrade, too. If not already doing so, another thing that might be really useful would be a wireless file transfer setup (a pro version... not basic WiFi). Transfer images to a computer while you're shooting, so that customers can view the results immediately.
Cat show photography is challenging. One photographer I know who specialized in it actually sets up a tent with studio lighting, a posing stand, various "toys" and such... then and shoots his subjects inside it. He sees fewer "escapees", as a result. Typically has a waiting line of customers, too.
PS: The only thing that I noticed in your sample image was the background... It looks as if you were using some sort of red backdrop, but it appears to not have been large enough, or this particular shot was done so that one corner of the image was off the backdrop. It would be easily fixed in Photoshop, of course.
Yes, of course there is similar to 35mm f/1.8 lens on DX, available for FX: A 50mm f/1.4 lens would behave very close in focal length... and 2/3 stop faster!
My acquaintance who photographs cats.. and has done so for about 35 years or more... was using a 24-70mm f/2.8 the vast majority of the time (on a DX/ crop camera). That zoom gives him rapid flexibility to deal with relatively unpredictable subjects. He finds f/2.8 plenty fast for the purpose, combined with studio lighting gear. In fact he stops down significantly most of the time, to get most of the cat well within the sharp depth-of field. His work has appeared frequently in Cat Fancy and other magazines and he has an extensive stock photo library... He travels to cat shows all over the U.S. Not sure if it's still the case, but in the past one of his top marketing "tools" was an annual calendar that he produces.... clients vie to get one of their cats included as one of the dozen or so featured in the calendar each year.
www.chanan.com is his web site, if you're interested.