Accurately Measuring Indoor/Outdoor Air Temperature & Relative Humidity
Looking at the source you provided in the comment, there is some other information you may be able to accumulate to help your study.


Control or Baseline.  You need an the same actual set of data from a facility in the area that is meeting the above-mentioned standards.  This needs to be as close to your target facility in size, structure and regional conditions as you can find so you can demonstrate that under the same Temperature and Humidity proper conditions can be maintained.
More Data.  You haven't mentioned Time in your question. Most building temperature studies show conditions throughout the day. The duration that the animals are exposed to extreme conditions may be relevant.  There are data-logging thermometer type devices available , or you may need to perform your tests once an hour during the day to collect enough relevant data to make your case.
More Data. You talk of taking measurements on an extreme day. Gather data on a variety of days, and you may find the care standards are exceeded more (or less) often then you expect.  This way you may find the lower limit of conditions which cause problems. Which brings us to...
More Data.  Once you have a measured example of the problem temperature and humidity reading, for a given day, you can collect data from local weather sources and you can present a graphic showing the number of days, or perhaps hours per year the animals are exposed to these conditions.


You mention Heat Index. According to NOAA,  Heat Index 


  is what the temperature feels like to the human body when relative
  humidity is combined with the air temperature... 


You might have to prove the relevance of this data to animals which do not react the same way to humidity. If you can find an external source supporting your use of that index, then by all means continue to use it. You may even be able to find a study showing that the animals feel the effects of humidity more then humans do. Look for, more data. The link above mentions a possible place to begin.

Personally I believe the main argument you will face would be the amount of time the proper conditions are exceeded. Like the people who leave the dog in the car, 'but it was just for a minute...'.

Good luck, no matter what results you find.