Well, if they had Odo show any changeling features within the Klingon ones, he'd never pass it off in front of the Klingons. That's the real reason. I just noticed how well defined his brow bones, cheek bones, nose, and mouth were as a Klingon, observing that this level of detail could have been given to him when converted back to normal. But... Odo chose not to, instead to retain the look he has been known for. I kind of wish he hadn't... and I'm just thinking that Rene would've enjoyed the break! I know it wouldn't have lasted more than a handful of episodes, but still...
I think the 'surgical alterations' in Star Trek are sort of ol' good latex mask (as we have seen René as 'Klingon' in TUC) or false Spock ears, but instead of latex they use something like medical artificial skin and flesh substitute (which looks more believable and natural), glued to the face with some special adhesive. And when you are done with your mission, you get a special hypo, so the adhesive loses its function and the make-up goes off. That's how I imagine it. In Odo's case, even the nose is not a problem, just add some ridges on it. The rest can be glued on too. Agree with Od0_ital- combining two make-ups would be too difficult to design. Klingon make-up (not the one from TUC) is a forehead with a wig, nose ridges and horrible false upper teeth. Founder make-up is a latex mask over all the face, and a blonde wig. The real life reason he can't do many facial expressions, is the latex mask, the Trek reason is, that he simply cannot do it. I do not know, how dr. Bashir could give him the temporal ability to do realistic face expressions, so I just will go with the explanation it's Auberjonois' real face.