The best I can recommend is the Beelink U59 for $170 after applying a $30 coupon. The reason is because it is the least expensive mini PC with a Celeron N5095 CPU (the best CPU in your price range) that also has two RAM slots. It comes with a single 8GB RAM stick and a 256GB SSD. When a 2nd 8GB RAM stick is installed, that will allow the RAM to run at full speed as opposed to only having 1 RAM stick. This can generally improve game performance by 5% to 20% depending on how sensitive the game is to RAM speed.

There are lower priced mini PC with a Celeron N5095 such as the Beelink Mini S for $139; it looks very similar to the U59. The primary drawback is that there is only a single RAM slot. That means even if you were to remove the 8GB RAM stick and replace it with a 16GB RAM stick, it will only operate at half speed.