Can I play Red's Challenge and fail the deck search?
No, because you are not searching for a specific type of card. It is public knowledge that your deck does have cards in it.
Source: Unbroken Bonds FAQ (2019-04-19), TPCi Rules Team (2019-05-02)

Relevant cards

Note: All applicable cards may not be listed below.
SM10_184
SM10_184
SM10_213
SM10_213