Well if you're getting a new CPU, you'll need a new motherboard to go with it and you'll need DDR4 RAM for your new motherboard, so you'll need to budget for that.
AMD CPUs are popular these days for their high core/thread counts and value pricing, but Intel CPUs are best for high clock speeds (both single and all-core) which makes them better suited for high fps games like MW.
As far as GPUs in that price range go, you're probably looking at a GTX 1660 super, maybe an RTX 2060 if you can find a good deal on it. I have a 2060 KO and I used to play a little MW on in conjunction with a 1080p G-sync monitor: I got roughly 80-110 fps with it on ultra settings.
You could replicate my performance with a 2060 and your current CPU, but lowering settings for higher fps will probably start bottlenecking your CPU.
The 1660 super could give you similar fps if you dropped settings further. It'll be a trade-off between eye candy vs fps. Obviously, the more powerful GPU you can afford, the less of a tradeoff you'll have to make.
If it were me personally, I'd put off the new CPU and motherboard for the time being and get a 2060 super and a good variable refresh rate 1080p monitor to go with it. I think that'll give you the biggest improvement per $ right now. If you want higher fps and quicker load times down the road, then you can upgrade your motherboard, CPU, RAM, drive, all that good stuff. Just a 1660 super by itself will still make a huge difference though.