The projector, if it is a roll out/put away piece of equipment, is a real question mark.

But, I would lean towards something like this:
https://www.projectorcentral.com/Panasonic-PT-VMW50U.htm

It's not as high resolution as other models, but it uses a laser light source and the 5,000 lumens is plenty bright enough for a 150" or so diagonal size on a 'off white' wall.

It has lens shift which can help get the image right on the wall, and it has a laser light engine, so it won't need regular bulb maintenance.

So, if it can be had for under $2,000, that's a pretty solid deal, especially long term, on a product that won't need any specialized standard maintenance or the cost of new lamps at $200+ on a regular basis.

To make this even cooler, the projector has a built in HDBT receiver, so a CAT-5 connection can be made directly from the extender to the projector (through a wall plate) and you will get the image from it.

Equipment list, using short links from the manufacturers sites. Most product should be bought used except for the TV, speakers, and projector.

TV: https://www.lg.com/us/tvs/lg-86UM8070PUA-4k-uhd-tv

Projector: https://na.panasonic.com/us/audio-video-solutions/projectors/portable/pt-vmw50u-3lcd-portable-laser-projector

TV Mount: https://www.monoprice.com/product?p_id=12278

Video Switcher: https://crestron.com/en-US/Products/Inactive/Discontinued/A-M/DMPS-300-C

Video scaling receiver: https://crestron.com/en-US/Products/Video/DigitalMedia-Endpoints/Receivers/DM-RMC-SCALER-C

Video transmitter wallplate: https://crestron.com/en-US/Products/Video/DigitalMedia-Endpoints/Transmitters/DM-TX-200-C-2G-B-T

Digital audio mixer: https://biamp.com/products/tesira-fixed-audio-dsp

Amp: https://www.ebay.com/itm/Crown-CTS1200-Two-Channel-1200W-Power-Amplifier-with-Cobranet-HiQnet-USP3-CN/401931087327?epid=14031748000&hash=item5d94f5addf:g:thsAAOSw0YxdnP7K

HDMI Cables: https://www.monoprice.com/product?p_id=24187

Cat-6 In-Wall Cabling: https://www.monoprice.com/category/cables/bulk-networking-cables/cat6-bulk-ethernet-cables
(Buy appropriate length and appropriate plenum requirement)

Speaker cable:
Non-Plenum: https://www.monoprice.com/product?p_id=2817
Plenum: https://www.monoprice.com/product?p_id=13716
Get 14 gauge in a length that covers the entire job. Probably 250 feet is enough. So, 14 gauge 2 conductor. (one 100'+ run to each speaker.

Speakers: https://www.qsc.com/live-sound/products/loudspeakers/passive-loudspeakers/e-series/e112/
(two of these)

Speaker mount kit: https://www.qsc.com/live-sound/products/loudspeakers/passive-loudspeakers/e-series/e112/e112-yoke-mount/
(two of these)

Middle Atlantic Locking Equipment Rack: https://www.middleatlantic.com/products/racks-enclosures/wall-mount-racks-cabinets/ewr-series-economical-sectional-wall-mount-rack/ewr-16-22sd.aspx

Please be aware with the equipment rack that there are DOZENS of similar makes/models which may offer far greater cost/value ratio on the used market. Check local Craigslist as well. I picked up an equipment rack for free that would have cost me several hundred dollars. Not sure of your exact location, but check around for deals on racks because they come in all shapes and sizes and if you pick up a rack without a front door for $100, then you may be able to just get the door (brand new) with a lock for $150, which still makes it a great deal. You just need to be sure the rack is 20"+ deep. It does not necessarily need to be wall mounted like the one in my link, unless you absolutely need it to be wall mounted.

Case (DiY) - $1,000

Programming/Installation: About $2,000. Knock it down some by DiYing things.

Considerations: How/where will cable be run? Will you need surface mount raceway to run cabling? Can you go above ceiling then drop cabling down from the ceiling to the TV/speaker locations to hide most cabling? In corners for speakers, you can use surface mount raceway (typically Panduit) to hide the speaker cable and protect it. To the TV, a larger single run will bring video into the TV and offer you a connection point for your laptop.

So, plan on there being some extra costs overall, but realistically, my budget of $10,000 really hits the nail on the head. If you need to buy a rolling cart for the projector, then plan on that as well. Little things can add up, so getting all the exact details perfect can take some time. Having flexibility in the budget, even if it is just putting in $500 or $1,000 for 'miscellaneous expenses' is a critical part of any job design and a standard practice in the industry. You will need wall plate covers ($3 each for nice ones). You will need an electrician to install power (no idea how much that'll cost!). You will need zip ties and other small installation items which just adds to the cost, but not singularly in any tremendous way. The TV will need to be properly wall mounted. So, proper hardware for that will be 10 or 20 bucks.

Plan for those intangibles.

Any projector that is purchased will be, with the intention of having it secured to the ceiling.

I took all of the products and placed them in a spreadsheet. However, for arguments sake...because I doubt the school will go shopping for used equipment. Would you mind linking products that are new. This way I can present them with both options.

Thanks

We (the school) received a second quote. It's moving in the right direction, but still way more then the district is willing to pay.