Is it possible to force-update Equifax that a specific credit card balance has been paid off?

From my experience credit card companies typically update your credit profile monthly. It would pretty much be impossible to call and "force" an employee to update your profile outside this schedule.

Why does it matter?

If the mortgage financing company wants to see a zero balance on this credit card, they you can provide proof via a screen shot of your credit card login.

It is unlikely to significantly change your credit score in the short term.

While it is likely that there may be some delay in the credit card company marking the payment and balances, the other significant delaying factor is CreditKarma (CK). While a real time poll of your score may show a value/state in time, CK doesn't update real time. Their updates lag by almost a month.

Likely because they poll all the information that they are getting from the bureaus and consolidating/collating info once a month. Depending on your billing date for a credit card, the update from that card may not make it at the time they poll but will make it in the next poll.

So CK's version of Equifax may not be the one that a merchant/entity may be seeing.

Also, from my experience, if your score is very good, then the excellent score doesn't provide you any additional bonus. If the merchant/entity/bank said that this is a reason for giving you a worse rate than you'd like, then that is odd. For mortgage, it is not just the score but income, debt-to-income and so many other factors that they look at. A score difference of, say 20/30, shouldn't adversely affect your application terms.