I can't imagine why you wouldn't use your B, especially since you paid for it

And you won't truly be able to ignore it- your hand will have to adjust to get around it uness you convert to a high C, and even then you will notice it.
Lastly, there are --and have have been for some time-- 5-string AUBs. You may end up wanting one if you keep at it. Since you've got a 5 now, I'd go with the supplementary exercises you're thinking of doing along with the ones in your method book.