Edited: Concerns were expressed about such items being suited for food use. I've added comments in the text on "Food Grade" items plus a note at the end.
How can I get a very thick or viscous paste (e.g. caramel, ganache, thick mayonnaise) into small-necked squeeze bottles without heating it up?
Consider using a "grease gun" as used by mechanics.
There are thick food grade syringes being made for that kind of application..
The plunger comes out at the back easily, so you can pour your sauce into the syringe, and then inject it into your dispensing bottle.
Similar to the other suggestions, you can use a large mouth water bottle, sports drink bottle, or restaurant style condiment bottle as a syringe. I've used a large mouth funnel into one of these with dry ingredients, then put the lid/cap back on to use as a squirt bottle.
If the bottle doesn't already have an opening, you can make one with a drill or knife. ...