@chuckr You definitely can leave the antenna outside in a plastic box. Your coax losses at 1.5GHz are pretty horrid. It's about 32dB loss per 100ft. for RG-316. So what you really want to do is to run power to a plastic box containing the antenna, the LNA/SAW filter, the SDR and the receiving computer (or any combination that does all of that). Seal the hole you make for the power supply well.
One of those snap-top lunch containers would probably work great.