"Yet we would like to find a way to show that income distributed over the months in 2019 on the P&L reports, so our monthly income appears more stable when compared to the previous year."

A Cash Basis entity got Donation Income in 2018. Period.

An Accrual Basis entity also got Donation Income in 2018, Period, unless there is a restriction on the use of the funds. If so, you would hold it, such as Grant Income that isn't yours to keep until you start spending it. Then it relieves the liability and it is Income at that point. Or, a Child care operation takes prepayments against next Semester, so Prepayments in Dec are liability until Tuition is charged for March.

I recommend the book Running QB for NonProfits, by Kathy Ivens.

"but this isn't really a liability, because we would never be required to refund it."

There you are, then. You recognize this is Income when you got it. What you need to do is help your Board read the Balance sheet, so that they understand their Equity position. And you set up Budgets. In other words, income in Any Year, is shifted to the year end before as "prepaid" and that simply is part of your Income Cycle.