And don't think that if you manage to get across the Mexican border with your firearm that you are home free. They have military check points, sometimes even in very remote and hard to get to places. I have never had a problem with them so long as you are polite and let them do their thing. They are mainly looking for drug runners. But if they find even cartridges in your vehicle, well then you look like you might be a drug runner to them. And just because you may be an American citizen doesn't mean your rights follow you to their country. You are guilty until proven innocent in Mexico, and you really don't want to spend time in one of their jails.
People do still go there to hunt, but you do need to book the hunt and let the outfitter arrange gun transportation and documentation.
As for the OPs original question, "what do you do". I'm fortunate that I have a sister who lives 20 miles from Tecate. I leave any weapons at her house if I want to cross the border. I know people who have been detained for just having a large belt knife on their person while walking in Ensenada. It didn't turn into anything, but they did have to go to the police station and explain that it was just a tool they carried all the time for camping tasks and they intended no harm to anybody. When in a foreign country it's best to be sensitive to their laws and sensibilities, rather than to try and force our "rights" upon them.