Can I configure the PostgreSQL locale so that it ignores umlauts in ILIKE queries?
A common solution to this problem is to have a "shadow" or "search" column for proper names. This solution has the advantage of being portable.

For example, here in Ireland, there are many people who are called MacGuire and there are those called McGuire. Depending on other names, some Irish people also use accents in a particular name and some don't. Furthermore, some people put a space between "Mac" and "Guire". And don't even get me started on the problems that names such as O'Malley cause with SQL, i.e. escaping the single quote.

So, in a shadow column, one might have rules
"Mac"/"Mc"/"Mac " -> MC (shadow columns are normally kept all caps - another source of varialbility (could have Mcguire). "O" and "O'" -> O.

What's more, say the first name was Rónán (Ronán and Ronan are also common), so another rule would be 

á -> A and 

ó -> O

You can apply these rules via ON INSERT/UPDATE triggers.

So, this would lead to a table thus:

Display_Name    Search_name    Address
------------    -----------    -------
Ronan MacGuire  RONAN MCGUIRE  1 Clontarf D1
Ronan McGuire   RONAN MCGUIRE  2 Malahide D2
Rónán Mcguire   RONAN MCGUIRE  3 Mary's terrace D3


And then the service agent will distinguish on the basis of address.

So, I am suggesting that something similar might be a solution to your problem.