Just my opinion ... I think the warranty is gone with the current pool since it was said it was due to mowing the lawn but I may be wrong. Interesting that a rock left a hole not only in the metal but the pool liner as well. If the company isn't offering to warrant their pool at this point I don't know what else they would do in the future.
If it were me I would drain the pool, pull the liner back in that location and repair the wall with sheet aluminum on the inside. I would first put some rust converter on it, paint it up to protect the converter and then put a patch on it. I would glue it to the wall and cover the patch with a couple of layers of duct tape. If the liner is holding water after the repair then all should be good there. As far as the outside, it may be possible to clean it up, maybe get some bondo (car body stuff) prime and prime then paint it. There's also putty type waterproof epoxy that may work. In order for the wall not to continue to rust you need to address any rust you see there.
I know draining the pool is something people don't want to do because of liner issues but I grew up taking down and putting up a pool for a number of years. If you prepare yourself for the inside work maybe it can get done in a few days after draining. If you don't touch the whole pool it may not be too bad reinstalling the area you took down. Once the patch is installed then it's just a simple matter of putting the liner back and filling the pool.