I can see from skyscanner and google flights that it is possible to fly CX and Latam to Lima. And I think CX also have codeshare with Latam. But why can’t I book using CX site?

Is there a way to book this itinerary so it will be in one PNR?

If I book them separately​​​​​, CX (HKG-LAX), LA (LAX-LIM), can I check through my bags to LIM?