Egg substitute for cookie-flavor syrup

You might consider going with pasteurized eggs (if you can find them whole) or a pasteurized egg product (a common US brand is "Egg Beaters").

Pasteurized eggs are eggs that have been pasteurized in order to reduce the risk of food-borne illness in dishes that are not cooked or are only lightly cooked. They may be sold as liquid egg products or pasteurized in the shell.

The 2013 United States Food and Drug Administration Food Code defines regular shell eggs as a potentially hazardous food, i.e., “a food that requires time/temperature control for safety (TCS) to limit pathogenic microorganism growth or toxin formation.

Because of the risk of food-borne illness caused by Salmonella bacteria that may be present in raw eggs, the U.S. Department of Agriculture requires a safe-handling advisory statement on all packages of raw shell eggs that are not treated to destroy Salmonella as follows: "Safe Handling Instructions: To prevent illness from bacteria: Keep eggs refrigerated, cook eggs until yolks are firm, and cook foods containing eggs thoroughly."

The FDA Food Code exempts pasteurized shell eggs from the definition of "time/temperature control for safe food” requirement to carry a safe handling advisory statement.

The U.S. Department of Agriculture also states, “In-shell pasteurized eggs may be used safely without cooking.”

Your local grocer may sell in-shell pasteurized eggs but you'll probably find it easier to simply by them out of the shell in a carton like this:

Note that Egg Beaters brand is made without yolks, so it probably doesn't have the flavor you want, so be sure to read the packaging of the different brands... your best hope of duplicating the flavor without risking added chemicals is probably to find whole eggs.

All that being said, I don't think it would be a good idea for you to attempt to store this for any extended period of time, at longest, you should abide by the "sell by" date on the pasteurized egg packaging.