Don't even bother asking the border folks. Officially, when you cross the border, you are supposed to go straight to the nearest border authority to report in. That means before you get the chance to do your climb. Then, after you reported in, you go to try to cross the border again, then have to report to the authorities, again, never actually reaching the goal that encouraged you to cross the border to begin with.
This is how it was before increased scrutiny brought about by 9/11.
Prior to 9/11, the only official way for you to cross without this Catch-22 was places with monitored crossings, like the Pacific Crest Trail and I imagine Hozomeen since there are rangers that patrol there.
I don't know what the official rule is now, but even those places might not be endorsed anymore. One thing seems consistant though, if you ask a border person they will likely tell you that you can't do it.