You have nothing to lose (other than phone time) from asking LH to rebook you onto the BA flight you prefer, although LH is likely to want to rebook you first onto a LH flight, then a LH Group flight (Swiss, Austrian, etc.), then some close partner such as UA, then other Star Alliance, and finally other (generally full service) carriers with which LH interlines, including BA.

How far apart are the times? Even if the earlier LH flight isn't ideal for you, I'd advise you to take it if LH won't rebook you onto the BA flight you want. With all of the delays and cancellations that are happening now, it's good to plan to allow a bit of extra time to travel if you need to be someplace that day.