You don't say where you live, but it looks like it could be a California zone 9. I know people love White Birch trees, but these are so disease/insect prone in dry summer climates, as well as needing a ton of irrigation to look good, and also deciduous in winter, so no privacy then. I wouldn't move the Lycianthes rantonettii to that corner unless it gets at least 6 hours of sun, or it won't bloom well. They are best in full all day sun for best bloom.

If you prefer to keep the birches, then by all means plant another, and just make sure you can keep this corner well watered. A compatible shrub for this corner might include Pittosporum tobira variegata, or also fragrant Osmanthus fragrans, or Edgeworthia chrysantha or Pieris forrestii, both could also look good in this spot with semishade or morning sun, but neither of these last two like hot sun and little water.

If you like the weeping character of the birches, but would prefer an evergreen, then a grove of Mayten trees could also work here, but this tree does have a tendency to send up shoots from the roots, which need to be kept trimmed occasionally. Acacia cognata is another fine textured weeping evergreen that does better in California conditions long term than Betula pendula, and wouldn't have the sticky aphid poop raining down on you like the birches do.

bahia: Yes I am in zone 9 CA. I forgot to put in intially and then was unable when editing so put instead in my subject line. Thank you for the suggestions for the shrub. We are aware that birch trees are decidious, but since they are already there for years (just have been grossly underwatered since we did not have irrigation in that corner and so had to handwater) we are going to keep them. Our landscape plan will definitely include the birch trees in either a drip or spray on a timer and they should look better (I hope). You are correct that corner does not get a lot of sun--we just need a fast growing shrub that will look good. I will look up the shrubs you suggested as that corner does get morning sun and then is more shady--though notice now the other tree was removed it gets more sun, but imagine that'll change once the birch trees get more regular watering. Our side neighbors also have 2-3 birch trees on the opposite side of the fence which are more mature--so between the two lots, it does in fact form a grove which we like (fence is being replaced and made shorter so should look better). I'm thinking of pairing them with a maple tree on the far right for additional shade/privacy from the side neighbors. I though the differing colors of foilage would complement each other.

bahia: We looked at all the plants you suggested. We are thinking of pairing the Acacia cognata with our two birch trees as it seems it is fast growing and will get the height we need without being overly large as it's evergreen. In this case, we would plant this in lieu of another birch tree--up on the raised bed behind the birch trees. What do you think?