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.
Same boat here, but unfortunately it seems like many manufacturers don’t consider the addition of a tape monitor loop which very few will actually use worthwhile. What I find really odd is many could easy have it as an option and the people who wanted it I am sure would pay for it. The electronics of an an in and out loop are really minuscule, but adding 4 or 8 more jacks to the panel is much more of an issue.
I recently purchased the Stellar Phono Preamp an exemplary piece. Before I bought it I asked Caleb to check to see if it is ok to run the unbalanced rca outs and the balanced xlr outs simultaneously. He checked with Darren who said that it was fine to do so. I run the rca outs to my NPC to utilize the A to D so I can record my records using Vinyl Studio. I have made recordings this way and the results are excellent.