The rules are ambiguous, but I think they lean to "yes."

As you know, it's not explicitly spelled out in the DMG. But I think they lean your way, and here's why:

1. On p.128 "Crafting a Magic Item" lays out the prerequisites for a character to craft a magic item: (a) a formula for the construction; (b) a spellcaster with spell slots--Magic Initiates need not apply--and knowledge of any required spell; (c) character level minimum.
2. Later, near the end of that section, we find a few paragraphs dealing with ancillary matters: lifestyle while crafting, custom items, and assistance. The paragraph on assistance calls out the character-level prerequisite for any assistants, but doesn't mention the other two. The designers could easily have said "if each of them meets the requirements to create the item," but instead mentions only requirement (c) from above. Admittedly, an argument from omission is pretty thin.

Unfortunately, there's a decent counter-argument of nearly the same strength: "anyone who assists is a co-creater, and thereby must meet all the creator's requirements. The level requirement is simply called out to make it clear that another spellcaster with the right spell but not of the right level could not assist." Of course, this line of reasoning would moot "if each of them meets the level prerequisite," which I find intolerable.

But it doesn't matter, because it's explicitly your call. (Of course, all things are ultimately your call, but this one's called out as such in the DMG.)

Back it up to the very beginning of "Crafting a Magic Item":

As an option, you can allow player characters to craft magic items. (loc. cit., emphasis mine)

Obviously we all understand the "golden rule" of GM-ery: Make decisions and adjudications that enhance the fun of the adventure when possible. (Repeated at the start of all of Wizards' published adventures, and a good paraphrasing of "The Dungeon Master" from the introduction to the DMG.)

But I'd suggest a broader understanding of published materials: what we purchase in the core set is not (just) a game. It's a system, with a game as a worked example included. The rules presented provide you with a default game playable out of the box, but also everything you need to construct a game after that fashion. Every rule, in my opinion, is preceded by a silent "and here's one way to do ___:"

There are many serious problems with this plan...

A lot of this is going into spoiler tags, because it reveals details of how things work behind the DM Screen.

First off....

It's almost certainly too heavy to Levitate, even while Reduced.

Levitate specifies that...

The spell can levitate a target that weighs up to 500 pounds.

The Reduce factor of Enlarge/Reduce...

The target's size is halved in all dimensions, and its weight is reduced to one-eighth of normal.

So, the maximum weight you can move with this combo is something that weighs 4,000 lbs.

The Maze Engine is described as

a 20-foot diameter sphere built of 1-foot wide bands of magically hardened and shaped bronze. Gaps in the bands show various gears and articulation arms within the sphere.

We will be charitable here. We will assume that the engine is about 75% hollow, accounting for the space between all the gears and arms and such.

Bronze has a density of 532 lbs per cubic foot. The Engine has a volume of 4,189 cubic feet.

So, going at 25% of the density of bronze (133 lbs/cubic foot)...

This thing weighs about 557,137 pounds. Or about 278 and a half tons.

Even if we assume that this thing is 90% hollow...we're still looking at it weighing 222,855 lbs.

Reducing the Engine would still leave it weighing 27,857 lbs. Far in excess of what Levitate can lift.

And, beyond even that...

This would take EXTREMELY precise timing.

The Maze Engine is sitting 40 feet above the lava. And is wedged there just enough that, once it starts running, it rattles its way loose and falls into the lava

if you cast Enlarge/Reduce on the Maze Engine...

its diameter just halved. It is no longer large enough to be wedged into that crevice and immediately falls.

The time it takes for something to fall a given distance can be computed with

$$ \sqrt{2\times\frac{height}{9.8}} $$

So, running the math with the numbers we have...

Once you cast Reduce on the Maze Engine, you have about one and a half seconds before it goes splash in the lava and is destroyed.

You probably should have all died from this, anyway.

The Maze Engine is NOT controllable.

Every time it goes off, it does something at random. About a third of these things are BAD. A few are TPK material.

If you guys got through this without all dying horribly, you are incredibly lucky.

You have a 4% chance of getting that ASI. Statistically speaking, to get a pair of ASIs, you'd have to set the engine off about 50 times. Once activated, it goes off every single turn until it is deactivated or destroyed (the game is not specific as to whether or not it can be re-activated)

During that time, each time it goes off it...

Summons 2 additional Mephits that you must fight. You have a 1% chance of all of your magic items being irrevocably destroyed. You have a 2% chance of being blasted back to the past again. You have a 4% chance of someone of the DM's choice being resurrected and appearing nearby (not necessarily friendly). You have a 5% chance to get blasted for 10d6 lightning damage. A 5% chance of getting attacked by a Green Slaad. A 4% chance of getting polymorphed into a flying snake. A 5% chance of getting hit with an AoE Disintegrate spell. And a 4% chance of being Petrified.

So, statistically speaking, in the amount of time it took you to get the two positive results you wanted...almost all of those things should have happened at least once, if not more.