There are two reasons to believe that less than 17% of illegally-present Mexicans entered using visas but overstayed.
First is the article Review of the Declining Numbers of Visa Overstays in the U.S. from 2000 to 2009 by Robert and John Warren. (Robert Warren was Director of Statistics at the INS, 1986 to 1995). According to Warren:
In Warren (1997), 17% of all unauthorized immigrants from Mexico, El Salvador, Guatemala, and Honduras were estimated to be overstays. For reasons related to the provisions of the Immigration Reform and Control Act of 1986 (a relatively higher proportion of EWI than overstays were legalized), this percent was higher than it would have been in the 2000-2009 decade.
[above "EWI" means "entered without inspection", the opposite of entering legally with a visa]
Secondly, in the GAO report in the OP (tables 6 and 7), the number of Mexicans who were overstays was analyzed by assessing whether or not Mexicans arrested in security operations were overstays or not.
Only 42/472 Mexicans or 9% of Mexicans arrested for being illegally present were found to be overstays.
Additionally, in every single security operation studied, when two or more Mexicans were arrested, it was always true (23 out of 23 operations) that less than half were overstays.
(The comment by @dsollen helps illustrate how BBC would come to the wrong conclusion. It's easy to look at numbers for all illegal immigrants and see a large fraction are due to overstays, and then misapply these statistics to Mexicans. But because Mexico shares a border with the United States, it is much easier for Mexicans to cross illicitly by land without a visa, compare to immigrants from Asia for example. So it is critical to look at statistics that are specific to Mexicans to answer this question.)