How to Find rate based on column header dates

I have a sheet with Charge Rates.
On this sheet I have a series of columns, each of which has a date as the header.
The rows have the names of people and their rates as of those dates.

I have another sheet which is the actual hours.
I have been trying to figure out a formula so that I automatically populate for each person what their rate is based on the name and the date the work is charged.

The problem is how to find which column is the correct column as the date of the work must be older than the column but not newer than the next column date.

I figured I would match to the closest value that is less, but didn't work.

So far the closest I have is :

=INDEX(Rates!B2:F4,MATCH(Invoice!B2,Rates!A2:A4,0),MATCH(Invoice!A2,Rates!B1:F1,1))