I don't know whether or not there exist pure (pdf)(La)TeX solution. But I had a similar problem (with bibtex) and fixed it by using the script below (Windows batch) to call bibtex
(if needed you could easily adapt it to biber) and provide it to TeXmaker in place of the native bibtex % (or biber %) command:
I had the same problem, I used the following workaround to avoid it
Then in my code, I use lstinlineProxy instead of lstinline. That way, texstudio is not aware of the usage of lstinline.