It very likely does not work by RAW, but it could be made slightly more reasonable
As others have noted, there are at least three obstacles to overcome to make this clever maneuver adhere to RAW: whether the bead of light can be considered an object, what constitutes as touching the bead (and may potentially cause it to detonate prematurely), and the weight restriction of the Catapult spell.
Is the bead of light an object
The target of the Catapult spell must be an object. However, the description of Delayed Blast Fireball does not specify the bead is an object, and while we'd usually consider any sort of bead an object, the description suggest it may just be a globule of light and no more. However, since the bead can be touched and subsequently thrown, having it obey the laws of gravity as an object with mass would, I believe that it is highly likely that the bead is an object that has non-negligible weight.
Is the Catapult spell touching the bead, and does it matter
One can argue that a Catapult spell does not touch anything, instead just magically propels it, but even that poses some problems when you look at it realistically.
The description of Delayed Blast Fireball clearly states: touch the bead and it explodes, either in your hand or at most 40ft away. In fact, it is important to note that the spell offers no other alternatives. It does not say "If you succeed on the saving throw, you may hold the bead to throw it on a later turn" or anything like that. And it makes sense if you consider what the Dexterity saving throw is supposed to represent. Saving throws in general are used to resist or avoid negative effects that would otherwise be inflicted upon you should you do nothing. In this case, you roll the Dexterity save to see if you are fast enough to throw the bead away from you before it blows up in your hand. The fact that you can throw it at someone you don't like is just an added bonus to keeping yourself intact.
However, there is likely a component of finesse, too. While throwing the bead away fast enough is imperative, you could technically do the same by hitting the bead with a bat, baseball style. But I'm sure we all agree that, if you did that, it would just blow up in your face instead (though, again, it is not specified in the spell description). Therefore, it is likely that a part of the Dexterity saving throw applies to carefully handling the bead as you grab and throw it.
What does that mean for the Catapult spell? Not good news.
While it's easy to argue that a spell which can chuck a 5lb object 90ft in a straight line surely makes the object fly fast enough to automatically succeed on that part of the Dexterity saving throw, the forces applied to the bead are likely too large to keep the bead intact. One could argue that, if a human can safely throw the bead 40ft away, that you can tone down your Catapult spell to do the same. However, that significantly reduces the effectiveness of the combo and delves too deep into uncharted waters to give a more concrete answer than "it's up to the DM".
Does the bead weigh at least 1lb, and does it matter
The only inkling of the weight of the bead comes from the fact that it can be thrown 40ft away. Let's compare that to a regular item, like Alchemist's fire (which weighs 1lb). Those can be thrown at a target using the improvised ranged attack rules: it can be thrown at a target 20ft away, or 60ft with disadvantage. The fact that throwing the bead at a target does not require a separate attack roll suggests that it is quite easy to throw, and likely much lighter than 1lb. Granted, you do not have to be too precise with it to do significant damage, but doing the same with a 1lb jug sounds rather unreasonable.
But what if we increased the weight? Naturally, you can't make the bead itself weigh more (unless you think the Enlarge spell would work), but you could put it in a flask and then catapult the whole thing. Or, to be more precise, bring the flask up to the levitating bead and carefully raise it open-end-up without touching the bead, so it ends up inside. Now, if you were to just catapult that, the flask would likely slam into the bead and have the same effect as the baseball bat from before. You would have to, again, carefully bring the bead to the inner edge of the flask, so the two objects accelerate as one as the Catapult spell takes effect. This would most likely require a Dexterity roll of some kind (possibly Sleight of Hand), which appropriately substitutes the Dexterity saving throw you would have to make if you just tried to throw the bead by hand.
If a DM would allow that, maybe they would also allow to fill the flask with liquid, saving you the trouble of finding a flask that is 1lb heavy when empty, and potentially softening the application of force to the bead, making the whole thing slightly more reasonable. Of course, the question stands whether immersing the bead in liquid counts as touching it. That one in turn raises a couple of other interesting questions: is casting the Delayed Blast Fireball spell safe to do during rain, or could a random raindrop spell your doom? How about casting it underwater?
Conclusion
Using clever tricks, there is a possibility of making the whole maneuver slightly more reasonable, but by RAW the problem of touching the bead is at least partially disruptive, and the problem of the bead's weight makes the maneuver impossible without said trickery. A DM might house-rule it to make it possible as is, or may require additional effort as described (to at least restore the requirement of a Dexterity saving throw), but they have more than enough arguments to just make it impossible from the start.
