The device manufacturer tends to be the authority on the repair of their products. If Acer is saying that the keyboard needs to be replaced, there's a good chance that there is no other option.
If both Linux and Windows show that the F keys are malfunctioning, then I also believe that the problem is hardware-related and the keyboard needs replacement.
Please keep in mind, however, that laptop repairs can get steep in price, and the cost of the repairs may end up being more than the laptop itself worth.