Yes, shoot that background first so you can match lighting on the GS stage. When shooting the background, there are sometimes 'happy accidents' that you can use to your advantage on the GS stage.
Example: While shooting the time lapse, a cloud rolled by, and drops an unexpected (but cool) thunder shower. Wouldn't it be fun to have the singer either get wet, or have someone step in and hold an umbrella over the singer?
Example: The trees are all bent over and flags flapping because it's a particularly windy day. So, why not blow some wind on the GS stage and toss a few leaves in there for good measure.
Example: The lighting conditions might change ... heck, let's say the sung goes down during the shoot. After the sun has set, you could turn on some lights on your location (car headlights). Then, in GS stage turn some similarly matched lights on your actor at the right-ish time. You can perfect the timing in post.
My point is, the background will offer you some ideas that may be cute, corny, or maybe even solid gold. Shoot it first, and see what if gives you. If you want to do something like the headlight gag, you can even shoot a few versions by going back to get it just right.
>>> EDIT: I just noticed I hijacked ConnerKent's sunset business. No offense intended, if I was seeking to steal anything from you, it'd be you BMI