I don't work with animals, apologies, but some points of may suit.

Free living TEE is only by DLW as mentioned above but is very expensive and technically challenging. If your animals are individually housed then there maybe some Doppler methods to determine spontaneous activity or its compensation in response to your exercise training protocol. Indirect calorimetry will give you confidence in your exercise protocol, ie you could see an increase in fat oxidation at rest or during exercise by 8 weeks of program. However, if you have the expertise and ethics then better to study histology/ enzyme systems of skeletal muscle before and at end of protocol in trained and un-trained groups. 

For accurate monitoring of total energy expenditure (EEtot), DLW or calorimeter is highly recommended. The method of DLW needs sophisticated equipment and facility (radiation control) and is expensive to set up. Even budget is available, it will take considerable effort to get it done right. Calorimeter (closed-circuit respirometer) is less expensive, but it only houses one animal at a time. Many calorimeters are needed if all animals are monitored at the same time.

If more than 30 animals in a group by randomization procedure and they live under the same situation (diet, ambient condition, etc.), I would argue that the effect of other activities is considered under control. Even there may be possible difference in the mean EEtot between groups; it would be minimal comparing to the much larger EE change associated with exercise training. In addition, if energy balance is a concern, the amount of food consumption can be measured by weighing method.