Hi Marius,
You sound like me - adverse to spending a disproportionate amount of money (relative to equipment) on cables.
When the BHA-1 just arrived, I conveniently connected the BHA-1 directly to my cheap NAD C542 CD Player. The headphones I use most frequently are the Hifiman HE-6. I was very happy with the sound.
I have since moved the BHA-1 to my main rig. In this main rig, my BCD-1 (SPDIF) and BDP-1 (AES/EBU) are connected to my BDA-1, which in turn is connected to my BP-26 Preamp. I then connected my BHA-1 to the Tape Out of the BP-26. I use a very inexpensive RCA cable (Belden cable; Canare RCA plugs) which I got a shop to custom-make for me locally for about US$50.
I am very happy with the sound I get. Despite using an inexpensive RCA cable, I am still able to hear "new" things on familiar recordings which previously eluded me. If you have a pair of RCA cables lying around unused - many of us do - perhaps you want to just try the Tape Out from the BP-26.