If you're at 1080p 144hz the 7700k shouldn't even be a consideration, it's a high end chip.
Short term, yes it will perform better than Ryzen at 1080p.
Long term when you move onto higher resolutions?
The 1700 performs identically above 1080p to the 7700k give or take a few frames either way, and with higher thread utilization coming into play with newer engines and DX12, it seems silly to invest MORE into a CPU that while temporarily may give you better performance, will fall behind in future.
I forgot to add this at the start, but @OP's budget isn't really suitable for either a 7700k or a 1700 anyway, so I think its more a question of will the 1600 work, since it is still a very highly capable CPU, and can easily be OC'd well.