I wanted it to be armored for rodent resistance just in case because it is an old building. I wanted it to be solid core, shielded and cat6a to ensure signal integrity. I wanted the keystone jacks to ensure that there is minimal movement to prevent failures from metal fatigue.

STP is really a waste of time and effort unless you really know you need it. Base-T Ethernet is already very resilient to EMI, and STP is fairly difficult to terminate correctly (you need a ground bonding system on both ends). Unless you are in a heavy industrial environment, it's not going to do anything more than add cost and frustration, and I simply wouldn't worry about signal integrity; if you terminate everything properly and are within the specs it will just work. Solid core is generally used for structured cabling (permanent installation), not patch cables, so it's not too surprising you can't find this premade. Unterminated, armoured Cat6 is readily available, but fairly expensive. Here's one easy source: https://www.amazon.com/Certicable-Armored-Outdoor-Direct-Burial/dp/B07YLYLSX2

It adds up to about $120. The others involved with the project are skeptical about the reliability of active components between the two endpoints, but I countered by saying that it is both immune to electrical interference and likely easier on the budget than a good copper cable.

They have reasonable concerns, especially if you buy uber-cheap, but this goes for anything you'd use to terminate the fibre if your equipment can't terminate it directly. How good is the power supply that comes with a $30 media converter, and how long will it last? What about the capacitors inside the box? And I share others' bad experiences with random failures of these sort of converters, though I really can't explain them, electrically they're much simpler than a switch but somehow seem to fail more. It will probably work just fine, but is it worth saving $100 over the Cat6 solution? Debatable.

I was under the impression that fiber optics are more reliable than copper over long for copper distances. What do others have to say about this?

< 100m isn't really a long distance. I haven't noticed any meaningful difference in reliability between the two over these distances for permanent installations. However fibre, especially single-mode is very sensitive to dust and dirt on the ferrule, and if you're re-patching it at all, it's going to be far less reliable overall. You also haven't budgeted for any cleaning equipment, which isn't exactly cheap by your standards, and really should be using it even on brand new cables/optics https://www.fs.com/products/39721.html . But in your case it's not plugged directly into the equipment so should be permanent, I would expect. Your media converters / switches will add points of failure. At the end of the day, in the situation you've described, I'd expect fibre to be less reliable.

Also, I did not want the signals being too strong, so I picked the lowest distance transceivers that I could find. Is there any risk of the cable needing optical attenuation given how short it is?

'Standard' 10km optics are designed to be connected back-to-back, so no attenuation will be required. If you'd like to know more, you can check the datasheet and compare the Max output power to the minimum receiver saturation power. In a 10km optic they should be the same, so the attenuation in the connectors and cable means you will always be safe. Some carriers do put attenuation to aim for a -10dBm power on the receiver to push it far away from saturation, but it's not necessary.

Lastly, I actually need a switch at the end where the streaming equipment is (which is a tough sell because of other team members’ skepticism of active components), but the budget is limited. Would it be terrible if I used this mikrotik device in place of the media converter and an unmanaged switch?

If you need a switch and you decide to go with fibre, much better to use this than the media converter. I'd put one at both ends, if you can stomach it at all.

Given the limited budget, are there any cheaper options with good reliability?

Standard UTP in conduit might not be cheaper in materials (in labour it's definitely more, but it sounds like that doesn't matter to you given your other choices here) but it's probably not much more expensive than either of the other options, and gives you options for future to pull fibre through for 10G or whatever comes up, and you can satisfy the desire to avoid active equipment.