I wouldn't exclude VA panels, especially compared with lower end IPS, VA can really show it's strength
screens I'd recommend are
BenQ VZ2470H (sub 200$)
Dell Ultrasharp (sub 250$)
Philips B-line 241B7QPTEB / 241B7QPJKEB / 241B7QPJEB (sub 300$, differences between those models is additional ports like mini-hdmi & number of USB 3 ports)
anything between 1-5ms is fine. unless you got jedi-like vision it's below your physical awareness anyway.
blb is a problem, but more with gamer focused brands like Acer or Asus
if you want the best picture, I suggest going for professional use focused screens like the ones I suggested
as for the games you're playing: pubg, cod "favour" faster panels, the others profit off the better looks of an IPS/VA panel
at 60Hz, I wouldn't gp for a TN panel though