budget i think is fine. just pick a good cpu (pick amd one which is good and cheaper).
U don't need a ultra hardware to work with 4k videos cause u can use proxy files and then just export: try looking youtube guide
You might consider Adobe Premiere Elements. It edits in 1080p without changing the original footage, then allows you to output the result in any resolution up to 4K. Note that this software does not use the video card, but almost entirely the CPU. My suggestion is to get the best CPU you can afford, then add a better video card later if you need it. With your budget, it may not be possible to edit in 4K, either in hardware or software.