Very common in real estate photography. I just use multiple flashes to get it right in camera - primarily because Realtors don't pay that much (expect free) and don't want to spend a lot of time in post. At most, if have reflection due to flash placement, will take additional with different light placement (or one turned off, eliminating the reflection) Since taken on a tripod and everything matches, layer the multiple prints and paint the good part of the lower layer forward. Other use HDR for the quick and easy, but that has issues with muddiness, clarity and color balance (i.e. HDR clown vomit). What you probably want to research is photo Fusion which is a combination of HDR type bracketed exposures with a flash exposure.
As an alternate, with that operating room above, would take a multi off camera indirect flash (bounce, softbox, umbrella - shoot through or bounce) photo, plus that ambient. Color balance between the two then blend the ambient in at about 15-30% opacity to bring some of the shadow detail back in, particularly the forward that the lighting would have destroyed the shadow detail.