Transferring non-operational vehicle title to new owner in different state

They have told you that you can not register the vehicle until it is operational. You ,yourself, acknowledge that you must be able to pass inspection in your question

but I live in Texas, specifically a county where registration depends on a safety and emissions check.

You work on the car and get it to the point where it passes their inspection. After you do that, you register the vehicle in Texas. If it takes you years to finish the car, you register it in years.

A title on a nonworking vehicle would do nothing, because the car can't drive down the road. A police officer can't stop a non-operational vehicle and demand to see it's registration.

Texas law states you must have a vehicle that will pass the inspection to get the title in your case:

Transportation Code Section 501.030 (a) Before a motor vehicle that was last registered or titled in another state or country may be titled in this state, the county assessor-collector shall verify that the vehicle has passed the inspections required by Chapter 548, as indicated in the Department of Public Safety's inspection database under Section 548.251, or that the owner has obtained an identification number inspection in accordance with department rule.

Not having a Texas title does not mean you do not own it. It means it is not considered a motor vehicle by the state. You still own it as you would own any other piece of metal you own. If you bought the car or it was given to you, you own it with or without a Texas title.

With or without the title someone wanting to take it would have to go to court and you could tell the court you own it and provide whatever proof you have either way, which is how it is intended to be done.