What size is your current monitor? 27"? 32"? And does it have adaptive sync / freesync? What model is it?
Also, you're spending 200€ for a new monitor - you could consider just getting a better GPU e.g. RX 5700 XT, subsidised by selling your old GPU.
If you need to save up a bit more first, then just try to put up with gaming at 1080p on a 1440p screen. I've done that once, briefly. It wasn't amazing but I kind of got used to it such that it was playable.
A 1440p monitor is more 'futureproof' for gaming, in that 1080p is a bit entry level now and where image quality is a factor, the extra resolution is a major advantage. In your position, I would find a way to make the monitor work and keep that, rather than trying to fit things to a GPU I will inevitably replace.
In the great majority of games, increasing the resolution puts much more onus on the GPU than the CPU. Therefore, if you are getting CPU-related FPS drops, a lower resolution monitor wouldn't help you anyway. However, you say that you drop the resolution to 1080p, and performance improves a lot. While there are ways you can confirm that, it's very likely you just need a new GPU, not a whole new system.