Dungeons and Dragons

You could give ambiguous riddles, and hint at the trick by having the creature get flustered and angry when someone provides an answer that is correct but not the expected answer.

Like "I am a key that opens no door."

There's monkey, donkey, turkey, piano key, musical key, whiskey, malarkey, lackey, jockey, computer key, keystone, typewriter key, and probably some more that they will think of.

There are a bunch of riddles that can be made more ambiguous by leaving out one of the lines. Getting the riddle wrong could also be a hint that the creature is not as clever as he thinks.

There's also the riddle my grandfather like to tell.

What is red, you hang it on a clothesline, and it has four legs?

A fish!

But a fish isn't red! (Well, you could paint it red and then it would be)

You don't hang fish on clotheslines (It's mine, I can do what I want with it)

Fish don't have four legs!! (Yeah, I threw that in there because I didn't want the answer to be too obvious.)

Make sure the players understand that it's not just you who is the idiot.

“You are driving a carriage. The carriage is empty when you begin your route. In the morning, you bring 2 merchants to an auction. In the afternoon, you take 3 farmers to the market. In the evening, you bring a noble to a ball. What color are the driver’s eyes?”

The creature was told this riddle by someone else, and misunderstood the answer: when the creature was “the driver,” the answer was “purple,” because its eyes are purple. It still believes the answer is “purple,” even though the players’ eyes are not purple.

“A man is sentenced to death. He has to choose from three rooms to receive his punishment. The first room has a firing squad holding fully charged Wands of Magic Missile. The second room is full of deadly poison. The third room is full of bulettes that haven’t eaten for six months. Which room should he choose?”

The creature believes the answer is the second room, because it is immune to poison. It does not understand that bulettes feed constantly, so a bulette that hasn’t eaten in 6 months would be dead, and refuses to grasp the answer no matter how many times it’s explained.

"Why is a Raven like a Writing Desk?" "Because they both start with 'R'."

This is Lewis Carroll's classic riddle the Mad Hatter asks to Alice in her titular Adventures in Wonderland. Interestingly, it is never answered in the story and was intended to be a nonsensical riddle that would have no answer. But Carroll did give a fun and clever answer later after years of people asking him. His answer was "Because it can produce a few notes, tho they are very flat; and it is never put with the wrong end in front!" Other writers have suggested clever meta answers as well, such as "Poe wrote on both," and "Because there is a 'b' in both and an 'n' in neither." I like to think that your fake sphinx would both be not very imaginative in his answer, and also clearly a poor speller.

"What walks on four legs in the morning, two legs midday, and three legs in the evening?" "A dog." (Because it always walks on 4 legs, which means it also is walking on 2 and 3 legs)

This is a twist on the classic riddle of the sphinx. The actual correct answer is "man"/"mankind".

Edit: Also, he could claim the answer is specifically "a dog" he will not accept any other 4 legged animal as an answer. If the players insist that, say, "a cat" or "a sphinx" are also valid answers for the exact same reasons, he could be really condescending and belittle them by insisting that they just don't get it. He could insist that it's understandable they don't get it because "It's a really tough one. A real thinker."

If a red house is made of red bricks and a yellow house is made with yellow bricks, then what will a green house be made of?

The right answer is glass but the bad answer is green bricks.

@DoctorTYVM
I did this last month with an idiot shiftless Hieracosphinx reluctantly guarding a treasure, and just had him ask the dumbest possible riddles in the most pompous and self-satisfied tone:

“Why did the chicken cross the road?”
To get to the other side.
“Which came first, the chicken or the egg?”
The egg. But he will argue.
“What is the difference between a duck?”
One of its legs are both the same.
“What's got two wings and is going to eat you?”
Point at the sky behind him and shout “That thing?!”

You've got a few options here, as I should think that ultimately the solution will be found by the not-sphynx's behaviours and mannerisms (though the questions themselves do help bring those behaviours out).

You could go for something everyone knows, like "What's lighter: a tonne of bricks or a tonne of feathers?" When the players get it right by saying they both weigh a tonne, have the not-sphynx insist on the obvious wrong answer instead (the feathers, as a single feather is lighter than a single brick). Then when the players explain why that's wrong and their answer is right, the not-sphynx pretends he knew that and was just testing them. Stuff like that.

If the campaign isn't super serious in tone, you could work-in famous pop-culture examples. Ask the questions for crossing the bridge from Monty Python "What's your name/quest/favourite colour?" With the not-sphynx not getting that the difficulty came from Tim alternating to a super hard third question for every other person he asked. You can even use the question about the African Swallow, with the not-sphynx not knowing if it should be laden or unladen, and just handwaving that one away when the players ask.

Ask the "What have I got in my pocket" question from the Hobbit. The players might try and answer "The One Ring" or something clever. When they exhaust their ideas and give up, the not-sphynx gets embarrassed and plays it down, admitting he forgot he doesn't have any pockets...

Cool idea overall though - I think your group will have a lot of fun.

So this creature likes to imagine himself a sphinx? I would just give simple riddles, and have it get frustrated if they say the riddle is too easy. Then hit them with a harder one and when they say that it's harder, have it be pleased.

I would say, use some classic riddles like the ones from The Hobbit, but then add one more line to the riddle that gives it away.

Also, recommend you have a look at this classic

The best worst riddle ever written:

"What's in my pocket?"

Honestly it sounds like the riddles may be clever, but the creature simply thinks the correct answer is wrong. Like "what has four legs in the morning, 2 legs in the afternoon, and 3 at night?" The answer is a person, but maybe the creature says it is another creature that operates literally on this premise. Or it presents the riddle about the head that lies and the head that speaks truth, and has a totally different idea on how the riddle is actually solved, much to the chagrin of your players who are, hopefully, smarter.

"Say my name and you break me. What am I?" "A child named 'nobody loves you'!"

The answer to this riddle is "silence", because saying the word "silence" will also break the silence. Obviously the less clever/funny version is about emotional harm to children.

By "bad" riddles, do you mean riddles that are so blatantly obvious that they're basically no-brainers? Or rather something like bad puns? Does the riddle itself need to end up in praise for the sphynx, in some "who is the goodest boy in the room" sytle?

How about "speak friend and enter" for those LotR fans in your group, where the answer is "friend" ?