"TP was advised to set up a corp with S-election"

Let's start with the issue of the advice in the first place, which is extremely misguided. You can search the web for why not to hold Real Estate in an S Corp. I would never take the advice of that person again.

"Should 50% of property income and expense be reported on each shareholder's 1040 Sch E, or 100% reported on the 1040 Sch E of the shareholder to which the property is deeded?"

Neither. No, you cannot just split the income and expenses for an S Corp and the deeded owner. That's because an S Corp is its own entity. Everything gets reported on the 1120-S and not the 1040 Sched E. Then, the shareholders are issued a K-1. You describe more how a partnership would work, but you have a corporation to deal with. The corporation is not a co-owner.

Yes, a ReFi often is required, because the lender needs the owner to be on the hook, and the owner would be the Corporation, as its own entity separate from the humans. That's the point of creating a corporation. That's part of why this was bad advice.

You did not state when this started. If 2021 was the first year, then you can deal with this a couple of ways. One is to execute a management agreement, where the S Corp is not the owner of the undeeded property, but will collect rental income and able to shoulder operating costs on behalf of the owner. That also means issuing a 1099-Misc Box 1 to the real owner for some amount as rental/lease income paid to him (bypassing payroll and the K-1 process as it would be part of expenses to the corporation). The real owner then has some costs and asset and depreciation and entries on the Sched E 1040 as the only owner.

And there should be some sort of legal framework, such as quitclaim deed or wrap around mortgage or something that is legal and common in their State for selling the property to the S Corp (which still is a bad idea).

Obviously, this is very late to be dealing with 2021. There have been a lot of errors and this is already mid-2022. You might want to get mentoring from a tax attorney to help deal with this, since it involves deeded/titled, and debt secured real estate.