These are great devices, I have one of these in my car. I does not broadcast an FM signal though, it is simply a modulator, not a transmitter. It would allow you to modulate an audio signal onto an FM carrier frequency which you could then connect to the FM Antenna input on a receiver, this could probably be distributed through coax cable and a couple of splitters. As a minimum it would allow a single FM radio device with an external antenna input to receive an audio signal from a single device with RCA style outputs. These devices are not regulated because they do not 'broadcast' their signal through the air, only within a wire.
FM transmitters are limited in power (and therefor range) by broadcast regulators (CRTC here, FCC in the US, etc..). Without a licence from a regulatory body FM transmitters are limited to the approx. 10' you've mentioned. I've used some that will do 20' on fresh batteries. These devices can sometimes be modified for higher broadcast power but you are then in violation of FCC/CRTC regulations. I've not seen an FM transmitter that does not run on batteries (I assume for transmit power purposes) because they are usually intended to be portable, like for your car.
Something within my expertise
Is it possible to do this within a core or a media director?
The way it works on a hotel system is thus:
You take any channel from a satellite/Cable receiver and you input it into a modulator(composite) and assign that modulator a channel number. You can do this with multiple Modulators and combiners (looks like a backwards splitter) in order to assign many receivers to create a full out channel lineup. The modulators transmit an RF signal through coax to your TV. You would now tune into that particular channel to receive what ever channel the receiver is set on. This type of system is very commonly found when modulating cameras into the cable signal that the cable TV companies provide.
BECAUSE it is a Closed caption signal, the strength of the signal running through coax doesn't need to be that strong so there wouldn't have to be that much worry about range. If there were a low quality signal the signal could be amplified at source to produce a better quality signal. Optimal signal to any device is 0db/mv (decibels/ milivolt). This signal strength can be measured using a RF signal strength meter.
Receiving channels on a TV tuner card would ensure a synced analog signal provided by the core.
Just a follow up on a solid Idea.