I reached out to some experts in this area, and conducted a little research myself.
First and foremost, it’s important to understand the laws in your state for handling asbestos. Legally, the uniform must be encapsulated. Asbestos is a highly regulated material, and by law, any exposed asbestos must be handled and encapsulated by someone certified and has completed appropriate OSHA training. Asbestos particles are hazardous, and if there is a risk that the asbestos components in the suit are broken, powdering, or crumbling, these invisible particles can become airborne. Proper Personal Protective Equipment (PPE) is key here.
The expert I reached out to reported that some institutions with known objects containing asbestos simply do not touch them. Others enclose them in sealed plastic, mylar or plastic boxes – using protocols established by industrial hygienists, or with help from an individual who has received the appropriate training or is state certified. Others dispose of the objects with the assistance from appropriately trained or certified individuals.
In doing my own research on this, I found a few sources that may be useful to you for background reading. I am breaking this part of my post into sections, because the site will not let me post more than a few links in one comment:
The National Parks Service has a great introduction to the identification, history, and proper handling of asbestos: https://www.nps.gov/museum/publications/conserveogram/02-11.pdf. This includes information about the PPE requirements for handling, as well as links and a bibliographic reference to other sources on asbestos.