"The basis for the agreement was the former member's transfer of his interest in the partnership to the long-standing managing member and his daughter,"

I think you are describing a sale of the interest.

"agreement obligates my client to make fixed monthly payments over a set time frame at an agreed-upon interest rate."

Which is an installment sale.

"Payments in 2020 were required, but none were made"

The entity cannot buy out the member with the entity's resources that would already be allocated to that member. The entity can be a source of funding for a buying member to use to buy out another member, but technically, this sale is between private parties. The entity didn't sell itself, take on a new partner or divest itself of a member.

"Anyone have any recommendations on the specific informational form and box to use?"

Nothing you described fits what you are thinking of doing.

It seems you have one person who sold to two others. There should be something documenting the leaving member's position and the agreement as Note Payable between the private parties.

You might want to seek a guru for some guidance on all of this. A mentor can make a difference.