I had to set up a web account with the IRS in the recent past and I just skipped the entire mobile validation by having them send the auth code to my mailing address. It arrived in two days and worked perfectly. YMMV.
I am, however, interested in what mobile carrier you use that had a "real" (not VOIP) mobile number and can receive SMS from "short codes", but did not work with the IRS ?
That's unexpected - basically every mobile number in the US is one of the big three or an MVNO operating with one of the big three networks ... can you share just a bit more about your setup ?
I think the IRS may have access to customer names associated with mobile phone accounts to confirm identity, but not all carriers have identity information for their customers, and I'd guess smaller carriers (or privacy focused carriers, if any exist) may not provide that access.