yes you can use the SFP port on your router using something like the finisar fclf8521p2btl active copper sfp module and running a copper ethernet cable to your streamer, but in my experience its better to insert a dedicated switch closer to your streamer rather than connect directly to your router. Routers are inherently "noisy" producing a lot of their own high-frequency noise as well acting as a hub for all the crap coming from other connected devices around your home network. The network switch positioned close to your streamer (there are plenty of relatively inexpensive switches offering SFP ports).
Obvs proof is in the hearing so try running straight from your router and compare the performance to doing the same but having switch in the chain using the same SFP connections. The sonic differences should hopefully be obvious.
Keeping your network chain as simple as possible is definitely the right direction to be heading in and avoiding fibre and other conversions is very good idea as these active processes generate and emit high-frequency interference picked up from their power supplies and internal powered processes.
Router > good lan cable > Dedicated network switch with uprated PSU (sfp out if you like) > your best lan cable > Streamer
My suggestion would be focus your spend on the switch not the router and get it as close to your hifi and as far from your router as possible and that SFP's are definitely worth experimenting with as long as your streamer is capable of taking a gigabit connection (some like naim are hardwired at 100mb/s).