Thanks, we try! Your problem is almost certainly due to a bad, missing or loose capacitor but hard to say if from Wii, the cable, the card or the monitor as you put it.
I own an L2 PVM that is compatible with the BKM-129X but since has R-Y/G-Pr/B-Pb BNC already, I haven't bought card or clone. I say capacitor because I had same issue with blue and red horizontal bars from SNES RGB:
https://imgur.com/a/0H9Y4pG
My first RGB-JP21 cable worked fine but is generic and I like name brand crap. Bought official SFC JP-21 cable and got said horizontal bars. I opened generic JP21 cable and saw 220 uF capacitors on the R, G and B lines. Measured 220 uF. The bad official cable has a lock that kept me from opening without damaging - and I doubted I could get refunded on damaging the cable. I instead measured Multi Out pin to JP21 connector on the G line and got a pF range. As in, capacitor died. Nice to use a pricy ESR meter that can tell you if capacitor is near end of life but I don't have.
See also RetroRGB Saturn page where a missing 220 uF capacitor + 470 ohm resistor to drop TTL sync to 75 ohm sync (that we don't need) causes blue horizontal bars:
https://www.retrorgb.com/saturn.html I'm not aware of SNES sync needing a capacitor but each system can be different.
Perhaps you know that 90s capacitors had issues and weren't built to last 25 years. Wii is newer but 90s CRTs, even our PVMs/BVMs, need capacitor maintenance too. Making your issue harder is I don't know if Wii itself is supposed to have the capacitors or the cable but can open Wii up and look and get cheap Multimeter that can measure capacitors. Check the card clone for any part that is loose. I don't think is issue but if you aren't using the video out, you should turn the terminate switches on to avoid adding video noise.
Rule out Wii by trying another, rule out cable by trying another, rule out card + PVM by trying RGB or Component from another system + cable pair OR use Wii + cable on another television. PS2 console + PS2/PS3 Component cables are cheap and plentiful. Early DVD players with Component video out are cheap. I was able to test my SNES on retro game shop TV with my Composite cable for free. LCDs and Plasmas as late as 2010 had Composite + Component inputs.
Obviously the last thing you want to do is open the BVM and mess with it but comes to that if you ruled out everything else. Plan would be to study the BVM's service manual and its mass of circuit diagrams and determine which capacitors are on the video line. Maybe someone has done the work for you and has video guide. Helpful that you checked Composite video being good since that rules out a lot of issues such as with the purity rings. In case you don't know, messing with CRT electronics is not a beginner task. I haven't done it.
