Earlier comments on NV might be a better direction than thermal given the cost limits. They can be dirt cheap these days and solve the "rabbits see the light and bolt" issue, though you don't get the heat signature helping find them like with thermal. It's not a direct answer to the cheap thermal question but it does address the problem.
You've also got the benefit that you can mount a cheap digital IR camera and screen to the back of your scope without needing to resight or have a gun that's limited to night shooting only without resighting with different optics. Digital IR is power hungry but cheap and can run in daylight as well as dark with an IR light source. If you're willing to learn a bit about electronics or modify things you can do a a lot of neat stuff. A lot of the rat shooters go this route - they're on a tight budget so have found cheap options for hunting relatively intelligent quick learning animals.
You can get torches in short wave IR suited for your night vision system, anywhere from 750nm to 950nm that are invisible to rabbits, deer, and us, but light the paddock up like it's day to the IR system.
Used a handheld Seek thermal unit and it was fine for finding rabbits - could even see their droppings from 20-30m away. Downside is that you're walking around in the dark ruining your night vision staring at a bright screen and basically spotlighting yourself. Ended up using my right eye for the thermal unit and scope and left for walking in the dark which isn't ideal, but was also nice being able to ID things without having to point the whole rifle at it to see it through the thermal like you end up having to do with a rifle mounted thermal unit.
For red light the red LED's make a big difference in output like MB says. I started with red tape over a light, then red filters, then eventually just went to red LED when clearing a big farm of rats... went from struggling for enough light to being well lit. The really bright red LED does seem to at least be a bit noticed, but seems like it's more confusing than actually noticed, so as long as you don't get them used to the idea that the odd almost visible light means humans, you're fine. In that case, switching to NIR doesn't give them any hints about what's going on.
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