Meshtastic

You sound like a good candidate for a router role, but just to be safe, I would generally avoid "router" unless coordinating with or organizing a coordinated community mesh. That would allow for the best placement without potentially adversely affecting the wider mesh. Client role will repeat just the same but without "overriding" other, potentially better positioned nodes.

The top of the mountain location, which would provide coverage for both sides, would be the ideal placement for a node in "router" role without consuming a hop to "jump" to the other side.

Stick with Client until you can coordinate and understand the problems with other modes.

That is all excellent advice. My only addition is to keep in mind that every new node extends the mesh. And every extension makes it more valuable for other to join. A little over a year ago I bought three nodes (for me, my wife, and a friend). At the time, we could only see our own nodes. Then we put up a repeater on a relatively high point in our neighborhood. Suddenly we could see several dozen nodes. Now we can see at least four other repeaters and around 200 nodes

You aren't likely to see that kind of grown in a rural area, but you are likely to see some. I think of it as similar to the early days of cell phones. You used to only get service in very limited areas, and never in the country. That has gradually changed to the point where coverage is more likely than not, even in remote areas. It got that way one tower at a time. The mesh is developing in the same way. And every repeater put up is a public service.

The advice given here is really great, and I agree with it. Stick with client since you can only see one side of the mountain. If you are able to get a node at the top of the mountain that can see both sides, then you can consider using router mode, but only after coordinating with anybody in the local area.