Travel Consent Letter from UK - Notary Public Required?

Hi all,
Has anyone had experience travelling with a child without the other parent - particularly when the parent travelling is not British and lives here on a Spouse/Family Visa.

-Married Couple
-Living Together (Joint ownership of house)
-Joint Parental responsibilities
-Wife is not British, travelling abroad with child (not to her home country)
-I, Husband will join them 17 days later (flights booked) for the last week.

In this scenario is it still highly recommended to get a notarised travel consent letter?

I have drafted up the letter and found a local Notary Public. The cost is £140 so while it's hardly a major cost it's something I'd prefer not to pay unless it's actually required vs me just writing up a letter, signing it, her travelling with childs birth certificate, marriage cert, home ownership recent document and bill, (she can even take my non-British passport) or any other document.

On the other hand- I don't want to risk their travel plans being disrupted as she will travel with her Mother who will then return home from the holiday destination 4-5 weeks later.

I think had she been British citizen I would just give the letter - but worried that her travelling on a foreign passport then there may be more risk of her being stopped and asked and then worst case being denied boarding.