Will there be any issues if BB sample rate is 44100, and my sound card rate is fixed at 48000?

There should not be on the Band-in-a-Box side of the equation. Band-in-a-Box automatically converts audio input to 44.1K sampling rate. You will have to find out how your sound card handles sample rate conversion.

I have a Creative Audigy sound card in my older desktop computer. The native sample rate for the Audigy sound card is 48K. The sound card driver has a setting to change the sample rate from 48K to 44.1K. Your sound card control panel may have the same option.