Are entry-level DSLRs compatible with Linux for transferring photos?

Most every DSLR/SLT/DSLM - certainly major brands like Canon, Nikon, Sony, Pentax - saves to SD or CF cards in a standard format, and additionally supports USB mass storage protocol, you DO NOT NEED any drivers installed on a host computer to be able to access the images taken.

Tethering, or upgrading camera firmware, might indeed prove difficult from a linux system.

Software that can process the JPEG format is plenty on linux systems.

What you do need, IF you choose to postprocess images from RAW format, is software that can understand the RAW format proprietary to the exact camera. Compatibility lists can be found for all the linux software (mind what version a linux distribution includes!) suitable for such purposes, in the online documentation for each software package.

If your intent is running a linux system as the camera firmware, there is no well known DSLR that supports that, though the Magic Lantern project for Canon offers something (augmented firmware) in that direction.

BTW, "under $1000" still leaves some Pentax options open, which might even be better value for money than other brands. Even more so if you consider buying used, near mint condition, from professional dealers ... there, several K3-I and KP secondhand bundles under $1000 by a well known online retailer... these are almost-professional cameras even, not "entry level".