There's a bit of a running joke among museum collections managers and conservators, that museums aren't heated for the objects, they're heated for the people. Museum objects (be they paper archives, paintings, clothing, photography, taxidermy etc), don't require heating, if anything, too much heating is detrimental. The cooler an object can be stored at the better generally (some stores, especially archival stores, are kept at around 16C). The main thing that museum collections require is stable, preferably low, humidity; it's often large and rapid changes in relative humidity that cause damage. Think of Scott's huts in Antarctica, a very cold, stable low-humidity environment - they have been preserved perfectly for over a century!
I would hazard a guess that the existing animal-displays could be retrofitted with museum-grade display cases, which would likely have built-in climate control (nothing spectacular, silica-gel for humidity control, and sealed edges to keep out pests is ideal). The low lighting in the building is also ideal to lower the risk of light-induced fading.