You're in for some decent levels of handwaving here.
So before we begin, let's take a better look at our goals: we want something that looks like this:
image link
To turn into something like this:
(art by Alexander Ostrowski)
The main problem one can see here is that the similarities end on having a beaked mouth, 4 limbs and 2 eyes. Protoceratops were heavy, fairly big (0.6 meters tall and 1.8 meters long) and their anatomy doesn't seem to indicate any kind of meaningful pressure for something other than a ground dwelling dinosaur that didn't really do any kind of hard climbing and had a mean bite (also fun little fact: we believe beaks evolved in birds due to being lighter overall when compared to teeth, but protoceratops over here had beaks and teeth). That makes a protoceratops with bird wings a little improbable. It's not that it couldn't evolve in any way, it's just that it's not all that believable, at least to me.
This isn't over though. Assuming that the protoceratops also had a respiratory system that made use of airsacks, which might not be too unlikely, since bird wings are a no go, we'll go for second best if you want something that looks like a griffin and first best if you want something as big as one: put your gloves and goggles on and prepare for some handwaving, because we're turning this:
Into this:
image source.
Not only are pterosaurs a much better alternative to griffin-like creatures due to being flying animals with hair-like structures called pycnofibers (which might actually be protofeathers and could indicate that sauropods simply lost these structures), but membranous wings are overall more efficient than feathered ones allowing the animal to grow larger (see how the biggest bird ever had a wingspan of almost 4 meters, while the biggest pterosaur had a wingspan of 10 to 11 meters)
So essentially, how do we make a protoceratops into a flying animal the size of a lion? I'd say one group of protoceratops would need to encounter the proper pressures to go down the following evolutionary line:
disclaimer: this assumes that your crazy scientists added protoceratops and velociraptor as the only vertebrate species in the world, since without plants everyone dies. For this specific line I'll also add in insects for the sake of making things easier (that is, more in line with what we believe happened). It also includes lots of handwaving to ensure everything happens exactly like we want it to.
they grow smaller (it will be already hard for a sauropod to undergo such drastic changes, and being the size of a sheep and weighting 80 kg as an adult doesn't help).
they move to more forested environments and become an arboreal species (not too unbelievable, the trees are essentially completely vacant).
they slowly move away from a herbivorous lifestyle, adapting to include insects and seeds in their diet.
as raptor species start to follow them in the trees, some species with rudimentary gliding abilities that can both escape from climbing raptors and ambush flying insects are selected.
the phalanxes on the 2 most outward digits of each front limb become elongated, supporting the patagium and the pectoral muscles become stronger, giving rise to a species that has wings similar to those of a pterosaur. The digits later come to fuse together as the hand undergoes more drastic adaptations.
At this point they're small flying creatures, that can hunt flying insects.
I do believe that the raptors could also evolve to become another group of flying animals, with the niches of flying insect eater available in other regions of the world, and here's how such a thing happening kicks the last stages forward:
the meeting between ptero-protoceratops and Veloci-birds leaves the former at a disadvantage, as their wings are pterosaur-like and not as maneuverable as those of birds (bats would be wrecking the competition here and that could be a problem, which is why I assumed a development of pterosaur wings)
being outcompeted in the niche of small flying insect eater, ptero-protoceratops undergo a new stage of selection focused on becoming larger without compromising the ability to fly. Their beaks take over as they loose their teeth and they undergo drastic changes to become larger. They pterosaur- like physique means they can grow larger than the velovi-birds as their wings can create more lift and their quad-lauching method of take off means they don't need their legs to be as heavy or as muscular as those of veloci-birds.
This is essentially the last problem, because in our world larger pterosaurs lost their tail as they offered little to no help in flight and functioned more like a dead weight than as an actual aid. The only way I see them keeping something that at least looks Ike a long tail would be if their tail pulled a Draco lizard move and develop a mostly reduced tail with elongated flexible rod-like bony structures that could spread apart and tuck together with muscles at the base, kind of like the following sketch:
-if everything went down in just the right way, we'd end with something pretty close to what we want: a toothless quadrupedal creature that can fly and looks almost nothing like a protoceratops. Some differences could include much longer limbs to what you might want and a shorter thoraxic region, as well as a long group of bony rods that function like the tail feathers of birds and a potentially longer neck and larger head (larger than the already proportionally large head of the protoceratops).
If everything goes down just right and assuming a best case scenario, we'd potentially end with something like this:
Don't judge this book by the cover though, this is basically a hatzegopteryx that had a tail with membranous rods attached to it. What's a hatzegopteryx? An azdarchid which was a little shorter than a giraffe, and had a reinforced beak and skeleton when compared to others of its genus. These adaptations lead us to believe that the lack of large theropod dinosaurs and island dwarfism which affected other creatures in its habitat allowed them to evolve into the niche of flying Apex predator that fed on mid-sized animals and could probably eat a human given their size. Don't take my word for it? How about Mark Witton's. Aka if by the time our Protoceratopterix came to be small and mid-sized species like protoceratops and velociraptors were still around, chances are that this thing would eat them (oh the turntables).
With our protoceratopterix completely done and much taller than a lion, let's compare them with some traits of an actual mythological griffin from the more classic design:
Has a beak - check.
Is likely carnivorous - check
Can fly - check
Has 6 limbs - not Check, you'd need some drastic mutations or a 6-limbed ancestor to get that.
as big as a lion - not check, it's taller.
lays eggs - check. Pterosaurs laid eggs, as did protoceratops, so it's unlikely that this trait would change.
is feathered - not quite. It will most likely have pycnofibers, but they aren't quite feathers and are likely more similar to strange hairs in appearance and texture.
warm blooded - probably check, flight demands a high metabolism, and you won't get that in a cold-blooded animal, plus sauropods were probably already on a mid-ground between being endothermic and exothermic.
So summing up, is it possible to make a griffin out of a protoceratops? It's very, very unlikely, and dropping a species of pterosaur probably would've allowed for a much smoother process with less obviously high levels of handwaving, but I'd say it's not completely impossible, especially since the herbivorous protoceratops and the carnivorous velociraptor are completely alone in a world filled to the brim with completely vacant niches they can occupy and convergent evolution exists (even if I'd say the existence of insects is very important to kick-start the process and create a niche of flying insect eater).
