My best guess from the context you've provided so far is that you're on the right track and it's 1Gbps 1000BASE-SX over 50µm multimode fiber thus having a 550m range, meaning the TRENDnet TEG-MGBSX is compatible. But it's hard to know for sure without reading the specs of that existing SFP module and the fiber optic cable.
There are lots of considerations in modern fiber optic Ethernet connections:
- Speed? (1G, 10G, 40G, 100G?)
- Ethernet protocol (IEEE 802.3 1000BASE-SX? 1000BASE-LX? 10GBASE-something? others?)
- Connector type (LC? SC? others?)
- Fiber type (single-mode? multi-mode? diameter?)
- Distance it needs to go (100m? 300m? 500m? 1km? 10km?)
Some of the choices on the list above constrain your other choices, but some are mix-and-match.
This is all to say that just because you see that an existing connection uses LC connectors in an SFP module, it tells you almost nothing about what flavor of Ethernet it's using, what kind of fiber it's using, what wavelength of light it's using, how fast it goes, or how far it can go.
I recommend you use the management capabilities of that switch to see what that SFP module identifies itself as. Or, if you can afford to bring that link down briefly, unplug the module and quickly take a picture of its label(s) and then plug it back in.
Yes, it matters what type of LC-connector SFP module you get. All of the ones that have distance ratings in kilometers are based on 1000BASE-LX over single-mode fiber, which is probably completely incompatible with what you're using for in-building links. Anything rated in hundreds of meters is 1000BASE-SX over multi-mode fiber. Ignore the 10 Gigabit stuff that requires SFP+ unless you can verify that your equipment has 10 Gigabit SFP+ receptacles (SFP+ is the 10 Gigabit version of SFP; if something is SFP without the plus, it only does 1Gbps, not 10Gbps).