If you can do that or not is up to the DM. Given the restriction (possession of the key's design) and the fact that it's easily "countered" by DM fiats requiring that the key has some specific property that the spell can't replicate, I can't see any problem with allowing it.
Trinkets are exemplified in pages 160 and 161 from PHB. One of the possible examples are "An Old Key". So certainly, by book's standard, a key can be a trinket. Can you make a functional key? In particular, does the spell allow you to insert such a level of detail that it actually works? It seems completely up to the DM.
"I wasn't sure if a DM would object" - ask him. Easiest solution ever.
As far as I know, a "Trinket" in English Language usually refers to jewelry, or, at best, "items of little value" - this was reinforced by googling "trinket meaning" to be sure. If we take that to face value, it would depend whether the DM considers the key is "an item of little value" or not. Again, it seems up to the DM.
Checking if allowing it is broken, fine or what
Usually, for most DMs I have played with, and certainly for me, "creative" uses of spell are fine, as long as they
- Don't overstep the purpose of other/higher-level spells.
- Don't create an incredibly easy solution for a challenge that was supposed to be hard.
The second bullet may seem harsh on the players, but I would rather not having them get access to that super important door that should require defeating the Dungeon's Boss by... using a cantrip.
Honestly, this can be "forced" by the DM while still allowing you to try, e.g. "the key needs to be made of diamond, because the door is special and breaks usual keys" - a diamond key is certainly not of little value. Or "the specific key is magical", w/e.
For other particular uses, personally, I would totally allow it as long as you were using it to have some fun (and not simply trying to break my game - okay, maybe breaking my game is your fun, but then we would be playing a very different game from what I usually DM).
Does it overstep other spells/features?
Not really sure. You might say it it crosses the boundary of Minor Conjuration, maybe? Other than that, it doesn't actually overstep, for example, a Rogue with Thieves' Tools using Lockpick, as you need the design of the key, which shouldn't be that easy to find, for things that actually matter.