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?

What about a piping bag? Fill the bag with a spatula, pipe into the bottle.

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.
These are used to pump high viscosity grease through small orifices.
The pressure can easily be as high as required. These are available new in the US$10-20 range and possibly even cheaper. (You can also pay far more, but there's no need to).

Grease guns come in many many many shapes and sizes, as can bee seen in these images.

Some GGs use a disposable cartridge, but some can be packed with grease, or other stuff, as required. A plunger allows the total contents to occupy only a small part of the tube. Even one intenmded for use with a cartridge could have a tube of the right diameter used inside as a cleanable refillable cartridge.

Food grade?: If there are concerns about food-grade quality of grease-guns then items such as this food grade grease grease-gun may be preferred.

"Food grade" has a very specific meaning. As long as the item does in fact meet regulatory standards then anything designed to be used in it, and the item itself, are suitable for use in contact with food*.

FOOD GRADE grease, grease gun

This mini GG seems to need no cartridge, only costs $10.60 on Amazon, and if not suitable could be used as a grease gun :-). Image here

This more conventional style gun for $14.72 on Amazon appears to allow cartridge free operation as an option.

*.*FOOD GRADE?: I'm in New Zealand. If something here is sold as "food grade" the authorities take the classification very seriously indeed. Whether this applies in any other given country is up to you to decide.

HOWEVER This applies equally to products sold for food only use. eg kitchen-ware with cadmium glazes (giving a nice cheery-kitchen red-orange colour) were sold for many years but are now banned and considered highly dangerous. Any item purchased from an unknown manufacturer without a proven record of caring and a reputation to protect should be at least examined for obvious 'issues'. And even items from 'reputable suppliers' sometimes fail the test.

I'd also be happy to use a suitably cleaned new item such as the Amazon $14.72 offering. Others may choose not to.

There are thick food grade syringes being made for that kind of application.. https://www.amazon.com/Syringes/dp/B07C71C1LH/ 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. If a homemade hole isn't good enough, you can plant it in the top of a small nozzle funnel that has a large enough top to hold the bottle mouth (without the lid) and squeeze from there. This can be messy and you'll need to keep pressure between the bottle and the funnel, which won't be easy.

On the other hand, you can get a condiment hand pump, instead. This would be on the order of those large, 1 gallon ketchup, mustard, and mayo containers that are available at grocery stores.

{Random pic from the internet.}

A wierd idea - if you squeeze the bottle, put its neck into the paste while squzeed and then let go, it will suck up the food inside? :o

I have a method, which I have used on salad dressing bottles in the past.

1. Take a bamboo kebab skewer (needs to be wooden, not steel, for the texture).
2. Put a funnel in the bottle's neck.
3. Pour in the thick liquid.
4. Put the skewer through the liquid and funnel hole to the bottom of the bottle, or as close as you can get.
5. Move the skewer up and down.

The liquid holds onto the skewer, allowing you to move it into the bottle as the skewer strikes the inner surfaces. It is fairly slow-going, though, so I would call it a last resort sort of solution.

An easy inexpensive method is to fill a plastic condiment bottle such as would be used for ketchup or mustard. Insert the tip into the narrower receiving bottle and squeeze product into receiving bottle or container. Seal finished container and pressure seal for safety.