Does food lose some of its mass in form of energy during metabolism?
Physics answer that is a bit silly

If you want to be real technical about physics, mass and energy are the same thing, so any chemical bond that has some energy and is broken results in a change of mass (mgkrebb's estimated in a comment the final mass would be 999.999999814 g, if you started with 1 kg of protein, also known to a very precise biologist as 1.000 kg).

This is sort of a "technically correct" answer that is totally silly from the standpoint of biology, but if your opponent wants to stand by it as "technically correct" based on physics, so be it. We are talking about changes in mass that are not measurable by modern science to sufficient precision, these are only theoretical ideas based on relationships between mass and energy that can be measured on a much larger scale.

Biology answer

Since this is Biology.SE, the biology answer is that mass is conserved in chemical reactions. If you greatly simplify the metabolism of, say, glucose, to the equation:

C6H12O6 + 6 O2 --> 6 CO2 + 6 H2O

you find that the net result is that the hydrogen and oxygen atoms become water, and the carbon atoms become part of carbon dioxide (yes I am simplifying where the actual atoms end up in the biochemical reactions, just talking in terms of net changes here). Other food molecules are processed similarly, and will produce some combination of water/carbon dioxide/nitrogenous wastes. If your cat eats 1 kg of food, that 1 kg of food will become 1 kg of something else out to many many decimal places: waste solids (including anything not digested at all) and liquids, exhaled carbon dioxide, lost skin/hair, etc.

Physics answer if you just want to win the argument

If you want to still win the argument with the physics-based answer that relies on mass-energy equivalence and ignores the fact that from the perspective of biology they are best understood as different concepts, then it's just as fair to go the other way and treat the lost energy as mass, in which case you'd consider the heat lost by the cat's metabolism as mass and include that tiny tiny tiny fraction in the 1 kg and call it a day.