That's going to be a tough one to find, never seen that exact combo.
The security-oriented coax to fiber units would NOT work for IP, those are basically analog to analog converters.
Your fiber switch doesn't have any copper ports, or GBIC ports for a copper adapter?
Comnet can do it. They have an SFP which converts to Ethernet over copper. It's used for extending Ethernet over UTP, but there's an RJ45 to BNC adapter, as well. Pair that with the appropriate fiber SFP, plug both into this SFP switch, and you're there. I haven't seen a two SFP port media converter, but they may be out there.
That being said, this is likely about an $800-900 USD proposition per end. It may be less expensive to get an EoC adapter and fiber media converter for each end, though it's multiple pieces of equipment.
Switch is a pure fiber, rather old switch. Thus my problem. We are discussing about using a new switch with SFP and use an Ethernet SFP on that end... The switch on the other side has Copper Giabit link so it is not much a problem ...
We have employed Comnet devices for this purpose. I think we also have another manufacturer's product as well doing extended Ethernet with PoE (about a 1500-foot run).
Several years ago it seems like we were using some IFS (or perhaps GE) coax to fiber converters. I cannot find one now, but I know we have analog cameras brought back to the head end via fiber.
Our Comnet devices have BNC connectors. Coax cable ends are relatively quick, easy, and inexpensive to swap to whatever you need. There are also adapters from F-conn to BNC, but I would recommend just changing the cable ends rather than using adapters. Probably would cost about the same, as the adapters' per-unit cost is usually a little higher than a regular connector.
For the Ethernet to fiber conversion from the Comnet unit, if your fiber switch uses SFP modules, you can use copper SFP modules. They are available in gigabit speeds. I have several in use on an IFS "fiber" switch.