back to the future - Why didn't doc and Marty just add a small metal plate or something to the old Delorean?

In the third Back to the future movie, Marty digs up a 70-year-old Delorean in a mine, and uses it to go back to 1885. Then, upon arrival, he gets his fuel tank pierced and it leaks all the fuel, trapping him in that time period.

Other questions have mentioned that they might have been able to siphon the fuel from the other Delorean waiting in the mine, but many reasons seem to prevent this course of actions.

But what about preventing the leak itself? They can see the hole the Arrow left, therefore know the exact location the Arrow would hit the car. They would only need to add a very small piece of metal on that exact spot to prevent the Arrow puncturing the car, and neither marty nor 1955-doc knows the new-model delorean enough to risk messing with it by removing that extra piece. They probably wouldn't even notice it. There is no paradox, as the 1955 team wouldn't do anything differently, the metal can be small enough to have no effect on anything (weight, aerodynamism, etc), so everything should play out the same, except for their own delorean to suddenly be full again.

BTTF already established its time travel rules with the possiblity to change in real time objects and events (Marty losing his hand in the first movie), therefore it would not be surprising to see fuel suddenly appearing in their own car.

Other options would also exist, like modifying the car to have a hidden backup fuel tank that automatically fills up with the main one, but isn't used, thus making them "find" this backup tank on their own car right away.

I'm mostly wondering if, and why, doc didn't mention any timey-wimey options, and went straight to the unlikely method of moving that fast with 100 years old tech.

He even invented long-burning chemical logs for the train, he should have at least tried something easier first in my opinion?

Answer

Why they didn't

Doc was at that point still very much against disturbing the timelines. He was stuck in 1885, but instead of asking Marty to come fetch him, he urged him to return to 1985 and destroy the time machine.
He was also adamant he would not fall in love in 1885.

He has hidden the DeLorean, which has worked, since Marty is now in 1885. Going back to the hidden DeLorean to add a piece of metal plating is exactly the sort of thing that could disturb a great deal, causing Marty to disappear from the timeline that landed him in 1885.

Whether it would have worked

It may not have worked. Remember that they didn't have that much time before Doc's date of death. As we've seen, the timeline does not change instantly. In the first film, it took a while for the McFly-children in the photograph to fade. Had they added a reinforcement to the DeLorean, the ripples in the timeline may not have been fast enough to reach them in time to prevent Doc's death.