I was thinking of using a single-mode LC-to-LC fiber with a RJ45-to-LC-SFP media converter,
A few things to be aware of.
The first is single mode verses multimode. Single mode fiber is usually cheaper and can support longer distances, but the transceivers/media converters are more expensive.
The second is, that unlike copper Ethernet where nearly everything is multi-speed, fiber stuff is often only single speed. You need to make sure all your components run at the same speed and use the same type of Ethernet.
The third is the physical construction of the cable. There are different types.
"patch cable" has separate reinforcements for each fiber. So the fibers can be separated without losing the reinforcement. It's usually used for short runs, for example from a switch to a patch panel, though I have seen longer runs of it in some datacenters. I wouldn't consider it ideal for outdoor use though.
Infrastructure cable on the other hand has reinforcement for the cable as a whole, but once the individual fibers are split out they have no reinforcement. There are many sub-types, armoured vs unarmoured, tight buffer vs loose tube and so-on, but for your application I don't think that matters much.
You probably don't want to get involved with terminating fibres yourself, so your best option is probably to buy it pre-terminated. Pre-terminated cables normally ship with a gland and a protection sleeve. The gland is attached to the reinforcement in the cable, and then the protection sleeve is screwed to the gland, protecting the individual fibers and providing a safe pull point.
Infrastructure cables usually come with at least four fibers, so you have some spares in the event of a fiber breaking.
You need to think about how you are going to protect the fibres after the protection sleeves are removed. In a datacenter this would typically be done by using a patch panel with a tray or box on the back to which the gland can be attatched.
For a single link in a temporary setup that is probably not the most practical setup, but you should still think about what you will screw the gland to, and how you protect the fibers between the gland and whatever they are plugged into.
The transmit side of one transceiver or media converter must be connected to the receive side of the other. If you are using duplex patch cables, this normally happens automatically. However if you are using infrastructure cable, you are likely to have to take account of this yourself.