Hi,
I pulled off the rubber shift lever boot on my TR4 and took a peek. It might be posible to get at the switch from above, if the shift lever were also removed. One long and one short bolt will get the lever out of the way. But watch in case the anti-rattle plunger and spring fall out of the hole near the bottom of the shift lever, when you lift it out. Then, with the right wrench, you *might* be able to get the reverse light switch out through the lever hole in the gearbox cover. It will probably be a lot trickier to get it back in, frankly!
On your car, I think you will need to remove the lower dash support yoke and piece surrounding the shift lever (TR4 just has the separate rubber boot). You've got nothing to looe trying this. I'm pretty certain you won't be able to get to it from below, and after you remove the lower dash support, there's not a lot more that needs to come out to go ahead and remove the gearbox cover, if necessary.
I agree with the other suggestion, check all the grounds first before assuming it's the switch. (Those original switches seem to last forever.)