I have Camera Raw 9.10.1 installed, Photoshop CC opens my .c2r files just fine. The compatibility page says I need LR 6.5 and Camera Raw 9.5+ but I don't know where the CC version of Lightroom falls on the version scale and opening "About Lightroom" just says "2015 Release, Camera Raw 9.0". System Info gives me Lightroom version: CC 2015 
Is this a limitation of Lightroom, or am I doing something wrong with the 9.10.1 installation? I'd rather not have to manually convert all my raw images before I can import and work with them in Lightroom.
My W10 updated version of LR shows "2015.10.1 Camera Raw 9.10.1". It appears you should update LR.
I just installed it last night (twice, just in case something had gone wrong the first time) from the Adobe CC launcher installer. Is there an alternate way to install it?
The problem is that your Creative Cloud application manager isn't updating Lightroom. You cannot update Lightroom by installing a new version of Camera Raw. Camera Raw is for Photoshop and for Photoshop Elements. Here is a link where you can download the update directly and install it. You will find the update under "Adobe Photoshop Lightroom 6" and it is for both Lightroom 6 and Lightroom CC. You will only need to install the latest update. Here is the link: