There is an extremely small reality that the model you are looking at will come anywhere near the light output that it is advertising. This is not a standardized US brand of projector, and DLP models often over advertise brightness as it is truly 'peak' lumens, center weighted, not a long-term true brightness.

For an actual 12:1 contrast ratio, you need to consider the screen size you are trying to achieve, and do a light measurement at the location where projection will occur.

In reality, if using short throw, a good ambient light rejecting short throw screen (ALRST screen) would be a near requirement in your situation. That will add a ton to the punch and effectiveness of any setup in a brighter room like you sound like you are dealing with.

Remember - no movie theater you've ever walked into has had 'LARGE WINDOWS' as a feature. This is basic reality. Light + projectors = bad image.

5,000 lumens sounds like a lot, but only if this is an official measurement as made by independent tests. Advertising from the manufacturer to that specification is often a lie, and moreso when it is coming from some non-US-based manufacturer. Chinese manufacturers love to lie about specifications. Epson has been suing them for their false claims. So, buyer beware.

Epson, who is one of the kings of short throw models, can only muster 4,000 lumens from their short throw laser...
https://epson.com/For-Work/Projectors/Classroom/PowerLite-700U-WUXGA-3LCD-Ultra-short-Throw-Laser-Display/p/V11H878520

I would go that direction over Xiaomi any day of the week, but still wouldn't expect much in a room such as you are describing.