Rich status information is available for Brother network printers e.g. at http://192.168.178.22/general/status.html How can I access status information of an USB printer?
By Brother utility?
https://support.brother.com/g/b/faqend.aspx?c=as_ot&lang=en&prod=hll9200cdw_us_as_cn&faqid=faq00003029_001
The Online User’s Guide of the HL-L2310D mentions only Mac and Windows Status Monitor: https://download.brother.com/welcome/doc100810/cv_hll2310d_uke_oug_c.pdf
Can you access web status page with USB connection?
Otherwise you need to write your own application/script.
The HL-L2350DW has a nice web interface. I discovered this accidentally when clicking on its device in the Fritz!Box router. I found no documentation of this feature. The LCD menu has an option to print 3 pages of settings.
Presumably the HL-L2310D also has the same print settings option. However the owner would prefer viewing the options in the web interface to printing 3 pages.
I have no idea whether the HL-L2310D also has a web interface and what its address is.