Fiber is not so expensive unless you are getting fleeced. Indeed, it can be cheaper than copper cabling. It's also the right solution to long runs.
Given the installation costs, it's likewise foolish to install 5e rather than 6A at this date and into the future for copper. The cost of the cable is a tiny fraction of the cost to get it installed correctly. With 6A your inter-switch links can run at 10Gb, even if your drops are only running at 1Gb. But your inter-switch links should really be fiber.
If you are stuck with a penny-wise, pound-foolish corporate mindset, drop in a switch every 100 meters or less. Throughput from [everything on a switch] is going to be limited by what the switch uplink speed is, to the extent that more than one thing on a switch wants to pass data up or down the uplink.
Commenting on topology without some sort of sketch of the building and knowledge of what other PWPF limitations on common sense are in effect is shooting in the dark.
Post diagram and switch model:
Is it more or less than 100 meters of cable between each of the 3 switches? Is it more or less than 100 meters of cable from each device to the switch it connects to? "The far end of the building" is not a factor here, unless there's a cable you are trying to run from this switch to that end.
If more than 100 meters, you need fiber to make the link. If less, copper is an option, but fiber is a superior option. If you limit yourself to that particular switch model, 4 fibers (2 links, aggregated) from each remote switch to the switch at the router is all you're going to be able to manage. Use 12 fiber cable anyway, it's not much more expensive if you shop well. A different switch at the router might offer you 8 SFPs and aggregating 4 links to the other switches, but changing all three to a switch model that supports SFP+ would be 2.5 times better than that...
Of course, if you don't actually have things like local file servers and this is all connected to a 50, 100 or 500 Mb (or even 1Gb) internet uplink, and nearly all network traffic is from or to the internet, not between local devices (or not with large data between local devices) this may not be a bottleneck in practice, as the internet connection will be the limiting factor. When looking to optimize a network I tend to try and eliminate bottlenecks on the assumption that if not now, at some point someone might hang a fileserver or database server off the network and it would be nice if the network worked well with that, but with a low-budget approach that may not be a current concern for your application - in which case a single gigabit link would be OK, though a dual aggregated link will offer minor protection from a cable disconnect. Connecting the distal switches with a normally STP blocked path will again offer minimal protection against one of them becoming disconnected, though in this layout a cable cut would be likely to take out both such links at once so the benefit is probably minor.
If cost is such a concern and your blue lines are fairly accurate, smaller switches might save quite a bit of money.
Going a bit further into why we push fiber: at 10 Gb, fiber transceivers are much more power-efficient than copper transceivers. Fiber is effectively immune to electrical noise, while copper is subject to it; the design of copper systems is intended to reduce such noise pickup, but that can be thrown off by pulling a cable too hard, bending it too tightly, or many other fiddly installation details. Fiber, particularly single-mode fiber, is effectively unlimited (itself) in terms of bandwidth; upgrades merely involve plugging in the newer, faster transceiver on each end, rather than replacing fibers in walls and ceilings. And, as mentioned, fiber is the only way to get past the 100m limit at full speed and without intermediate devices. But in your case, you seem to have missed that the "100m limit" applies to "cables between devices" not "overall size of the building" - when you get to a switch and go out another port, you start counting again at 0m from the switch port.