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? This is a problem I've had with several types of food/sauce. 

I'll make a big batch of something like a thick caramel sauce or an extra-thick mayonnaise and it simply won't pass through a funnel into a squeeze bottle.

I want to be able to transfer the sauce (or whatever) into a narrow-necked squeeze bottle so it's easier to decant. Even though it's very thick, it'll still come out of a bottle under pressure.

Is there some kind of in-line pump or similar method which I could use to pump these thicker liquids from a bowl into small-necked bottles?

I know such pumps exist for industrial applications but I want something usable at home. 

(My only idea was to use a sausage stuffer, fill it up with the sauce, and apply pressure with the pusher from the top to dispense it from the end like sausage meat, but the pusher would need to be air-tight and I don't know which ones are or not.)
What about a piping bag?  Fill the bag with a spatula, pipe into the bottle.