No, they do not need to be warm blooded.
Whale sharks average 9.8m long and 9000kg. They are cold blooded, and regulate their temperature by swimming close to the surface. This is demonstration that it is not at all necessary to be warm blooded to attain great size in an ocean environment.
Different exotherms have different tolerances for cold weather (and fully exothermic fish in particular can be found in very cold environments). This indicates that there are adaptations available to reduce the effects and/or likelihood of brumation occurring. I expect your sea-going reptile takes advantage of some of these if it is exothermic.
However, there will be advantages and disadvantages to being warm or cold blooded that may make it more or less likely your reptiles are cold blooded.
Pros and cons of exothermy
The one really big positive for exothermy is that it is very efficient in its use of resources. By being exothermic, your lizard would be able to go very long periods between feeding. This would be very useful if it is hunting comparatively rare big targets like whales and sharks. Kills would be few and far between, so it would need to be able to tolerate long periods of starvation.
The cons for your reptile would probably be reduced endurance and/or speed, and reduced ability to function in cold environments (arctic seas, or deep diving). These make chasing high-endurance, cold-water adapted and deep-diving whales somewhat tricky. They would need to ambush whales in order to make a successful kill.
On the diving front, being very large would be a benefit to your reptiles. If they are sufficiently insulated, they could maintain their bodies at surface temperature for quite some time while diving as the bigger they get the more mass they have compared to surface area.
Pros and cons of endothermy
Being able to regulate one's temperature flips these over. You can sustain activity for far longer and can survive in the coldest of climates with sufficient insulation. However, it comes at a big cost in energy requirements.
Again, being large helps with heat retention, but they will still be using quite a bit of their energy intake to regulate their temperature. This would make them better at hunting whales and sharks, but they would have to catch a lot more of them and more regularly than otherwise. This is an issue as whales and sharks tend to be at the top of their respective food chains so are not typically as numerous as the smaller fish and crustaceans they prey on.
The middle-ground: Mesothermy
Exotherms are animals that have no internal method of maintaining their temperature. Endotherms are animals that can regulate their temperature using internal metabolic processes. Mesotherms are animals that are able to somewhat regulate their temperature through other methods.
Great White Sharks utilise the heat generated by their muscles to maintain a temperature higher than the surrounding water. Your cold-blooded reptiles would benefit greatly from a similar adaptation as it would allow them to better pursue whales and sharks while retaining the efficiency of exothermy.
So what's more likely?
You have a few likely options here.
The first is an exothermic oceanic roamer (or localised species) that hunts by ambush. It will go a long time between kills, and may well be a scavenger as well as actively hunting. Catching big prey will have a high failure rate, but it is able to sustain prolonged periods of starvation. It will also have a restricted range to warmer tropical waters, or the surface of warmer temperate seas.
The second is a mesothermic oceanic roamer that is able to hunt better by endurance, and able to survive in much colder seas. In reality, it's not too far different from an air-breathing great white shark.
The third is an endothermic localised species. It is more efficient than the others at hunting whales and sharks, but is unable to sustain itself in the vast, empty expanses of the open ocean. It will be found where there are high concentrations of its prey species. It may also migrate to follow different populations of its prey species, or remain in one place and prey on a number of different species (say, if there is an area that whales use to give birth and nurse their young).
tl;dr They can viably be endothermic, exothermic or mesothermic, but these will change its behaviour and ecology. Of these, the mesothermic option has the closest real-world analogue so is probably the most likely.