If it were me, I’d keep the XLR cables connected to the BHK and run a set of RCA to XLR cables to the ADC. I doubt you’d notice any significant degradation in sound quality. You ADC and recording would more than likely by your limiting factor here. I’d run a good quality RCA and use an RCA to XLR adapter - Silnote makes a very good one. Actually, I’ll bet Silnote would make you a very high quality custom RCA to XLR cable along the lines of $300. I have some of their XRL cables and a shielded phono cable and they are fantastic. Just email them if you’re interested and they’ll get back to you quickly. They’ll even burn the cables in for you.
Here’s the adapter:
Just a thought.