I don't know if any of this is helpful...

"But I'm not sure the 8606 "proves" anything"

Line 2 has basis as dollars. If the taxpayer was taken RMD, then every year has the updated amount.

"and then allocating the basis by the percentage each institution"

By account. You don't use % now. Basis is a fixed $ amount. If that account is split two ways, then basis is split 50/50. If there are two accounts, it's still all done by $ for basis per account (not institution). It's literally, "How much they paid in."

https://www.napa-net.org/news-info/daily-news/case-week-aggregating-rmds-0

"there are special RMD “aggregation rules” that apply to individuals with multiple IRAs. Under the IRA RMD rules, IRA owners can independently calculate the RMDs that are due from each IRA they own directly (including savings incentive match plan for employees (SIMPLE IRAs, simplified employee pension (SEP) IRAs and traditional IRAs), total the amounts, and take the aggregate RMD amount from an IRA (or IRAs) of their choosing that they own directly (Treasury Regulation 1.408-8, Q&A 9).

RMDs from inherited IRAs that an individual holds as a beneficiary of the same decedent may be distributed under these rules for aggregation, considering only those IRAs owned as a beneficiary of the same decedent."

Normally, you can't combine inherited IRA, but since these are from the same owner, they can be combined for each beneficiary. So, your beneficiaries can combine the various accounts they inherit from this person, from different institutions.

As noted, the basis stays with the account. Pub 590B covers this. The section is:

"IRA with basis"

I would also read the section on Miscellaneous Rules for RMD, Separate Accounts, to see if this action should be taken.