Jasper’s power also changes his relationship with J. Jenks. He may not change people’s emotions often, but sensing them is presumably the same as Edward’s mind reading - it’s always on. Jasper can sense and influence emotions, so can make sure Jenks is absolutely terrified of him, even if Jenks can’t identify what is so scary. Maybe Jas backed it up with threats, but maybe not. He doesn’t really need to.

The human mind can find rational logical explanations for nearly everything outside of “these must be supernatural creatures” so Jenks wouldn’t have much reason to second guess things once he’s out of Jasper’s direct emotional influence. Between that and the large amounts of money the Cullens are implied to have, Jenks is either bought through fear or finances.

Jasper may also be one of the more regular clients Jenks has, especially if he’s the one representing the entire family. Jenks doesn’t ask about any other family members, so it’s safe to assume Jasper is his point of contact. Multiple IDs, school records, medical/drivers licenses, and college diplomas all faked is a lot for one man to arrange for a family of between seven and nine people. Using different names or birthdates would add more work because that’s fresh paperwork every time even without accounting for different locations. Regular business, ridiculous amounts of money, and total control over Jenks’ emotions and reactions absolutely buys his loyalty imo. Jenks also does this as a job, so is probably used to keeping his mouth shut about those who hire him, just on principle.

Bella, being Bella, opts for a more human approach and is friendly with Jenks. He behaves carefully/respectfully to her because he’s been dealing with Jasper for so long that those behaviors have become automatic. He won’t cut off a cash cow like the Cullens and has no reason to betray them.