There are some points to clarify first:
- A « UV map » is a 2D projection of a 3D mesh. This is what you can see and manipulate in Blender's UV Editor and it's what's define how any 2D texture is applied onto your 3D model.
- When you create a picture file out of that UV map, it's called a « UV layout », and really is only a picture of your UV map, not the UV map itself. It's like cooking your meal and taking a picture of your finished plate, it's only the photo of your plate, you won't eat that (I hope) but you might use it somewhere else.
I recommend using those words, to avoid confusion.
The reason to export a UV layout is usually in order to use it as a reference to create a picture file in some external software which suits your UV mapping.
It's necessary only if you need to take parts of a picture (or several pictures), move them around and fit them into pieces of UV. If you just want to apply a picture file without the need of such editing, you just have to plug your picture file in your shader.
The resolution of your UV layout doesn't really matter, it just has to be precise enough for you to see the limits of each UV island. You don't have to adapt your final picture resolution to your UV layout.
What does matter is the resolution of the picture you want to use in your shader. But also the ratio. As long as possible, use square pictures. Non-square pictures usually create unwanted deformations as the UV mapping format is a square and will conform any picture you plug in.
What you could do in your case, is either copy/pasting parts of your initial picture under UV islands as you want and export the resulting picture alone. Or take your 2960x3000px picture and cut it down to 2960x2960px and save it as is.