Will they cover the fee for the accountant to fix their own mistake?

If the IRS position is determined by the Tax Court to not be substantially justified, you can petition for fees for your representation. That’s a higher standard than merely winning your case; if you win the case, but the Court finds the IRS position was substantially justified, you’re on the hook for your fees.

In my case, they say they mailed a check that never arrived (direct deposit is not available).

They have a process for exactly this.

https://www.irs.gov/faqs/irs-procedures/refund-inquiries

If the check was not cashed, you can get it canceled and a new one mailed.

And yes, I've dealt with the IRS a few times some of which they owed me money. It is a bureaucracy, but the process will work even if it takes a bit of time.

yep, done that a couple of times. No response, no progress, nothing. FWIW, my wife did get them on the phone once and she was told we needed to prove we never received it -- that was years ago now.