SartellPeds,
I am not using AC cloud based version or 2x remote desktop for chrome. So I am not quite sure. If I understand you correctly you want to print using your chromebook remoted into your desktop running AC in the cloud.
Are you using any local desktops on your local network behind the firewall (intranet) to run on site printers in your office?
If so, the you should be able to hook them up going to this site...
https://support.google.com/cloudprint/answer/1686197?hl=enFor any PHI I use only the local printers behind the firewall (intranet). For teaching materials and non PHI I will print on my local printers using Google Cloud on occasion if I am not remoted into my desktop with VNC viewer (intranet) option.