Does estimating OIRFs from an SVAR (with uncorrelated errors) mean we are orthogonalizing the error structure twice?
Usually if you are estimating a SVAR model you would use structural impulse response functions (SIRF) not orthogonalized impulse response functions.  SIRF are not exactly the same as just IRF or OIRFs. SIRF already takes the identification problem into account when the SVAR is estimated.
To learn more about this you can have look at this blog about impulse response functions in R - the blog is brief but explains difference between simple impulse response, orthogonalized impulse response, structural impulse response and generalized impulse response. More info can be found in sources cited therein.