You could take a composite (analog) output (usually a RCA yellow female jack), feed it into a DVR (RCA to BNC adapter needed) and if that DVR has a video stream out using its LAN IP then BI should be able to use it.
Many older TV's have a composite input but finding a composite output might be the challenge. If so, look for a digital TV signal receiver with a composite output and feed that to the DVR. Those were sold in the early 2000's when TV stations were switching over to digital and many older TV's could only tune in analog. I bought one about 2004, so not sure if they're still sold; may be pretty scarce, though.
EDIT: here's a digital tuner with composite output (to the DVR) and HDMI output (to a monitor) for $30 =>
Digital Converter Box for tv, 1080P tv Tuner Box with hdmi Output, ATSC TV Digital Tuner, TV Recording&Playback, USB Media Player, Timer Setting Function Set-top Box Digital Channels Free