I think your theory is flawed because of the Bohr effect. CO2 in your blood increases the affinity of oxygen for hemoglobin. Higher oxygen and less CO2 means that most oxygen can't combine with hemoglobin and just ends up oxidizing somewhere which is definitely not ideal. You can easily test this by hyperventilating, which despite taking in much more oxygen and breathing out much more CO2 will cause less oxygen to reach your brain until you pass out.
The only way to breathe out more CO2 and lose weight would be to make more CO2 by increasing your metabolism, for example by taking thyroid hormone. This brings us full circle to your theory as it shows that increased CO2 breathing isn't the cause of weight loss, but a consequence of a faster metabolism which will in turn burn more calories and create weight loss.
The only way to breathe out more CO2 and lose weight would be to make more CO2 by increasing your metabolism, for example by taking thyroid hormone. This brings us full circle to your theory as it shows that increased CO2 breathing isn't the cause of weight loss, but a consequence of a faster metabolism which will in turn burn more calories and create weight loss.