In search of small portable projector for gaming

Hi folks

I am in search of a small portable projector that can be connected to my son's PS5, that would mainly be used for gaming along with the occasional movie. We don't have a screen, so it would be projected onto the wall. The projector would probably be positioned approx. 2 metres from the wall in question. My Facebook timeline has recently been flooded with suggested products - see link below with one example:

https://www.poplawillow.com/products/mini-projector---4k-fhd-1080p-supported-2-2?utm_source=fb&utm_medium=Others&utm_campaign=120213959949770180&utm_term=120213959949820180&utm_content=120213959949640180&fbclid=IwY2xjawG0KYNleHRuA2FlbQIxMAABHaoUAZVJm6ldc985mPZsLNxjTwJQ0kZY_T1L4Mj7JJ5I2JeZd3N1jfuzXA_aem_ND7K5IJI91TZ2GND2XEbFw

I would prefer to buy a product like this from a real shop so that I could check the quality in terms of picture and sound. However, any of the leading shops which have simialr products in stock cost in the region of €600, whereas the products on Facebook seem to be costing less than €100. I'm not willing to spend anymore than €100, but would be nervous of buying products that pop up on Facebook.

Can any of you guys provide some advice? Have you already got one of these products at home?

Thanks a lot!