I'm trying to find a word to describe a restaurant space that is the opposite of cozy. The only word I can think of is cavernous, however I don't wish to give the sense that the space is particularly large. It's simply that it is not cozy.
I think OP has a slightly distorted understanding of the word cosy - it's an almost accidental connotation that cosy places are often quite small.
The key sense is comfortable, warm, sheltered, intimate/friendly, which also normally implies soft lighting, plush furnishing, relaxing background music, etc.
When referring to a "non-cosy" restaurant, it all depends on whether you want to be positive...
bright, spacious, lively, airy, bustling, popular, buzzing, flamboyant, etc.,
brash, flashy, loud, garish, gaudy, noisy, etc.
"Cozy" usually implies "small". If you want an opposite that does not imply "large" I would assume that the space is still small, but without the positive connotations of "cozy". I would suggest "claustrophobic". If that's too extreme, maybe "closet-like", or "cramped" would do? If you want to sound almost positive, you might use the phrase "space efficient", though it might not work for your context.
I am a cozy, cuddly person. When I am in certain warehouse or industrial spaces, if they are not filled with the expected "sound and vision" of artistic, or warm, usually organic touches of an eatery, I can get depressed fast. It's almost a feeling of agoraphobia: cold, sterile, empty, disconnected, detached, remote—like there is not enough love, light, creativity, happiness or human connection to fill the space. (In restaurants, usually the wine, food, and temperature take care of that immediately!)