That's what I was thinking; however, it's pretty unlikely anybody wil ldo this unless they REALLY know you.
The problem is, it's potentially putting their ass on the line for something they have no control over or anything to do with.

Lets say I come to you and ask you to do this for me. I get s bill saying you installed the motor for me, at $400 cost.
Great, right?
Now, a week later, I show up at your door. My motor isn't running right. Or even better, my WIFe shows up w/c it's her car. Now I have a piece of paper saying you did teh work, so I could ask you to fix it. If I were a jackazz, I've got all the proof I need to file a lawsuit against you.

Or, better yet, lets say I don't do that, but file w/ the warranty folks that sold me the motor. This is the "right" thing to do, right? Guess what, I'd bet they'll be calling your shop to follow up on it, look into the mistakes you made and who knows, pushing you for reimbursement due to their incurred costs. Eitehr way, hot water for teh shop that "did" the work...

Here's an option... find out what exactly they need to know about the "installing" profession and their "company". Just a name and a bill on official looking letterhead? Will they actually need a TIN for them or anything?
In many states, setting up a "company" is actually quite easy, esp if you don't actually charge or make any $$.
Just file for a business license and make up a business, "Cool Toyota Guy" or whatever..
Now make up some letterheasd, and a bill that statest he work, and billed not for cash but "traded for in-kind service".
It always comes back to teh definition of "professional"....

Sounds like fraud to me.

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