I have a Toyota gear box, so my '60 BN7 is a bit different. Early on I decided to replace the defective speedo cable, purchased one that was a bit long, and found that the existing cable was "trapped" behind the extension panel you describe. As I soon had reason to take the transmission cover out of the car, I chose to drill a hole through the transmission tunnel just behind the extension panel you describe. Looking from the driver's seat at the circular top portion of the transmission cover, I made my hole at about 11 a.m. This was large enough to pass the cable through beneath the carpet and then up to the instrument. I padded the hole a bit, of course, and now can simply pull the cable in and out without removing either the tranny cover or the extension panel. There is often the question of the Toyota right angle drive that directs the cable forward from where it plugs into the transmission. Since those drives are expensive and reportedly trouble prone, I chose to buy a longer cable that enabled me to use a very gentle curve on the cable (and housing) as it went up and over the transmission and out through the hole I described earlier. The challenge I had was fitting the cable end bit that inserts into the transmission, basically the drive tab. I had to do it twice because I failed to first connect the instrument end of the cable to determine the final length of the internal drive cable. I made it too long, and thus I could not tighten down the cable housing and one end or the others. So I did it twice, getting it correct on the second try.
So, how did that work? No problem whatsoever for several years until recently. I am saying the speedo ran smoothly with no bounce. However, after I disassembled my speedo in an effort to improve its function, upon reassembly I found the magnetic piece that drives the bit that transmits the cable rotation to the instrument's needle was no longer turning that vital part. No explanation for why that happened. I then replaced original the magnetic bit from an old speedo, and then found my speedo worked but had the bounce familiar to those who experience cable problems. I have no idea why or if changing the magnetic piece prompted the speedo needle to start bobbing when the car was driven at a consistent pace. I did try adding lubricant to the cable and attempted to reposition the cable in the hope that it would remove the bounce. No success. I also tried another method suggested by a veteran Healey owner, backing off the threaded coupling at the speedo a half turn or so. Didn't work. You must be a masochist to work on these cars. I'm considering buying another long cable