I'm not sure what context Google provides in those text messages, but if it is just a one-time code; how would I know, you aren't trying to log into one of my accounts?
This is literally why I said it would sound suspicious. Most people wouldn't be able to tell. Which is why I said you'd need to provide some kind of sufficient proof of your identity (or some adequate alternative) to address their fears.
Fantastic suggestion! Definitely a great option if they're willing to take the risk.
If not, another one I was thinking was asking them to meet at their local police department and looping in an officer into the story or something. Different things work for different folks so they'd have to get creative to find the right solution.
I would be very wary of associating multiple Google accounts together. You don't need a single mistake from your personal account to cause a closure of your business account - and Google is known to do that. I recently had "nicer" auto_corrected to "ni66er" (Android keyboard swiping) even though I've never used that racial slur in my life. I could see how that might set off a chain reaction leading to account closure.