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.