Per what you've cited from the DMG, a spell cast from a magic item will be cast at the lowest possible spell level unless the item's description says otherwise.
This is also the way ritual casting works - you expend no spell slot, which means that the spell cannot be cast using a higher level spell slot.
... but not this time.
Remember that "unless the item's description says otherwise" bit?
Mizzium Apparatus says otherwise (emphasis mine):
In addition, you can attempt to cast a spell that you do not know or have prepared. The spell you choose must be on your class's spell list and of a level for which you have a spell slot, and you must provide the spell's components.
You expend a spell slot to cast the spell as normal, but before resolving it you must make an Intelligence (Arcana) check. The DC is 10 + twice the level of the spell slot you expend to cast the spell.
So in this case, the specifics of the Apparatus requiring you to spend spell slots override the general case of magic items not requiring you to spend spell slots.
Given that you're expending a spell slot, you can just spend a higher spell slot than you need, and if you succeed on the Arcana check, you'll cast the spell you chose with the higher-level spell slot.
Now, if you fail the check, the table you roll on to determine what spell you cast instead will be dependent on the level of spell slot you spent on it, not the native level of the spell you tried to cast, which will lead to a similar 'no upcasting' effect... unless you use a spell slot of 6th or higher level:
If the slot is 6th level or higher, roll on the table for 5th-level spells.
The slot is still at least 6th level, even if you're using it to cast a 5th level spell, and so upcasting effects may apply as normal.