Welcome to the forum!
I'm not very good with printing issues. I started to answer this message, then I got distracted with something else. And meanwhile, Lazur has posted. His information is likely more proper than mine. But I'll go ahead and post anyway, in case it helps.
From what I've heard (and observed), Inkscape doesn't have very sophisticated printing support. Often I read that some people recommend saving as PDF, and using PDF printing options.
Although having said that, when I click Print from the Inkscape window, I get a print dialog with 2 tabs. The first tab is General, where you can choose which kind of printer to use, and some other options. The other tab, called Rendering, gives 2 choices - Vector and Bitmap. If I choose Bitmap, a DPI spinbox becomes activated. This is with version 0.92.2.
As far as I know, to do what you want, you could export a PNG at 300 DPI, and then print to the particular size that you need. (Inkscape doesn't offer to print at a certain size. I'm guessing if you have a printer connected to the computer, it will give you the option. But I don't have a printer connected, so I'm not positive.
Which version of Inkscape are you using? If it's an older version, then it sounds like you could either upgrade, or at least temporarily install the current stable version (to see these same options as I see in the Print dialog.