The International Chewing Gum Association tells us:
Gum base is what gives chewing gum its “chew.” It is made of a combination of food-grade polymers, waxes, and softeners that give it the texture desired by consumers and enable it to effectively deliver sweetness, flavor, and various other benefits, including dental benefits.
A more complete description of what makes chewing gum sticky states:
Terence Cosgrove, Professor of Chemistry at the University of Bristol and Chief Scientific Officer of Revolymer, explains that the chemical bonds between the molecules in a polymer-based chewing gum make it difficult to remove from a surface. When you pull a piece of used gum off a surface, most of the energy goes into stretching the polymer bonds in the gum, rather than actually breaking the bonds between the surface and the gum. Since polymers are long chains of covalently bonded molecules that give the gum its elasticity and "chewiness," the attractive forces between the atoms are formed by the sharing of electrons . Thus, the bonds between the repeating units tend to remain intact as they lengthen and contract from an applied external force.
The temperature at which the polymer base is exposed to the air also affects both the elasticity and adhesiveness of the chewing gum. When a piece of gum is heated by saliva and deformed by the grinding of human teeth, its polymer chains align in the direction of these forces. The degree of alignment is a function of the magnitude of applied stress, which explains why the gum becomes tougher and less elastic the longer or more rigorously you chew . After the gum is removed from your mouth and placed in a cooler environment, the drop in temperature causes the orientation of the polymer chain to freeze, resulting in a hardened piece of used gum.
Most commercial polymer gum bases are hydrophobic (water-insoluble), which is the reason they stick easily to oily surfaces and are difficult to remove, even with the help of household cleaning solutions. A hydrophobic substance tends to repel polarized molecules like water and attract non-polar compounds, such as the grease and grime on streets. A hydrophilic substance, on the other hand, behaves in the opposite manner by attracting water and repelling fats and oils . A compound that exhibits both hydrophobic and hydrophilic traits is known as "amphiphilic" . These substances, which attract both water and oil to some degree, can be used to create synthetic polymers that could help eliminate the sticky mess of chewing gum pollution.
Note: The numbered references in the above block quote can be pulled from the linked article. The article also details efforts aimed at producing a new type of chewing gum that doesn't stick to things as much as normal/currently-manufactured ones seem to. It's a good read.
But to your main point, I've linked the MSDS for a popular product for removing gum and related compounds (it's called Goof Off), which consists almost entirely of acetone (also commonly found in nail polish remover). Given the deleterious health effects of acetone (and its high vapor pressure), it's not a good candidate for your needs.
Something non-stick and more permanent, such as a Teflon spray might be ideal for your needs.