I'm in favor of just going with a Class die and being done with it. It's simple, consistent (Thieves and Fighters already have it), and establishes a marked differentiation between class checks on non-class checks.
I am firmly of the opinion that if you don't have something in your background that suggests you know something about a check you can't do it. It's not in your narrative. Sure, from a simulationist point of view everyone can attempt something but DCC isn't that game. You should only do things that are appropriate for your character and it is the Judge's responsibility to make sure that you aren't blatantly put in a situation that is deliberately solved by resources/abilities you couldn't possibly have.
If you must have a "non-class" check die, then make it a d16 - which is consistent with a secondary action die. d16 still allows for a chance to tackle a DC15 check and if you are allowing a check in the first place there should be a chance.
I think the point of skill checks was meant to allow for actions that aren't covered elsewhere. Most dramatically interesting situations should be handled with Saving Throws or class abilities. Thief abilities I think muddy the water, which is probably why they were made to be % abilities to separate them from the rest of the game.
It's interesting to me that you have a nearly complete set of mechanics but then a hole for "skills".
Why not change the "saving throw" concept and make it a ability check concept. Fort save bonuses could apply to all STR & CON checks. Reflex save bonuses apply to all AGI and some INT checks. Willpower apply to all PER and some INT checks.
From the beta skill page:
Breaking down doors, bending bars, and lifting gates: Strength
Climbing: Strength or Agility, as appropriate. A sheer wall uses Strength; a craggy cliff or tree uses Agility.
Searching and spotting: Intelligence
So, change that:
Balancing = Reflex + AGI
Feats of Strength = Fortitude + STR
Resist Poison/Disease = Fortitude + CON
Resist physical control = Fortitude + CON
Dodge/Avoid Trap = Reflex + AGI
Perception = Reflexes + INT
Resist compulsion = Willpower + PER
Resist mental control = Willpower + PER
See through illusion = Willpower + INT
Sneaking = Reflex + AGI
Never add Class Dice to "Resistance Rolls/Saving Throws"
Fighters/Dwarves add Class Die to all Fortitude checks
Elves/Wizards add Class Die to all checks involving magic
Clerics add Class die to all checks involving the divine
Elves/Halflings/Thieves add Class Die to all Reflex checks
All characters add Class Die to background knowledge checks
If you consider reflexes to also include awareness, this works pretty well. You could rename them to Body/Mind/Reflex to better capture that feel.
This gives you a reduced consistent level bonus to reflect overall experience and basic class differences.
I also like the idea that Luck modifies everything.
EXAMPLE: Crossing a rickety bridge while under attack by goblin archers. Bridge is 60' across. DC 20 Reflex (Agility/Balance) to cross the bridge at normal move (and full AC); DC 15 at half move (-1 AC); DC 10 at third move (-2 AC); DC 5 is quarter movement (-4 AC)... roll and apply the result that matches the roll. All characters are 3rd level and assume a 10 AGI. Warrior would have a +1 to his roll, Wizard +1 as well, and the Elf is +1 with a Class Die of d5, while the Thief would be +2 and add his Class Die of d5. The Thief has a fair shot at the DC 15 target, and may burn Luck to do the DC 20 and make it across in a single go (double move), while the others are likely to be crossing at the DC 10 level and thus have to make 2 more rolls. The Elf will likely get closer to the DC 15 and cross in 2 rounds.
Perhaps that is a little too mechanical, but that seems dramatic especially if they are getting attacked. If they aren't getting attacked then the rolls aren't all that necessary and while a rickety bridge is interesting color, how one crosses it is simply a matter of story and not dice.