As others have said, an enterprise product is definitely called for.
As an academic exercise only, to do 250x 4MP @ 15 FPS
with Blue Iris you would need a
minimum of four servers, with most or all of the cameras configured to use sub streams or limit decoding (or else the CPU load would be way too high). Sub stream support is getting pretty decent in Blue Iris but it is still annoying at times when for reasons unknown the main stream takes 10+ seconds to load in when you maximize a camera. To work without sub streams or limit decoding, you're looking at needing at least
ten i9-9900K or faster systems and they would all be running a very heavy load (25 cameras or 1500 MP/s each). Two servers is a pain. Four is more than I'd recommend to anyone. Ten is just "lol". Even if you got them all running stable (which is no guarantee), it would quickly become apparent how lacking BI is in terms of management features to accommodate this scale.
By the way, that is an impressive home server room.