Calculate a simple linear regression and get values a couple of values. Try 1 + 2x. Then take the exponent of the values. That's what the values should be. Now try calculating the 1 + exp(2)x.

In R it would be:

exp( 1 + 2 * 1:3)
# versus
1 + exp(2) * 1:3

It should be pretty obvious these won't be equal. Given that it should be clear that you don't even back transform the beta coefficient let alone make a CI around it.

What you can do is generate the linear function and CI around that function within the transform space and back transform those final values.

Calculate a simple linear regression and get values a couple of values. Try 1 + 2x. Then take the exponent of the values. That's what the values should be. Now try calculating the 1 + exp(2)x.

In R it would be:

exp( 1 + 2 * 1:3)
# versus
1 + exp(2) * 1:3

It should be pretty obvious these won't be equal. Given that finding it should be clear that you don't even back transform the beta coefficient let alone make a CI around it. The back transformed coefficient is useless.

What you can do is use the equation to generate points in the linear function and a CI around that function within the transform space and then back transform those final values.