You could count up a number of successes
You can set a success cutoff number for each pool size, and just count all rolls equal or higher than that cutoff as successes and demand a given number of successes per pool as high or higher. This gets you around summing, you just look up success die.
To be honest, I think this is a horrible experience, because you essentially need to have a table at hand to remember the cutoffs, and it is not very exact either but here you go:
| Pool | Cutoff | Successes Needed (at least) | Probability |
|---|---|---|---|
| 1d12 | 12 | 1 | 8.33% |
| 2d12 | 10 | 2 | 6.25% |
| 3d12 | 9 | 3 | 3.7% |
| 4d12 | 10 | 3 | 5.08% |
| 5d12 | 12 | 2 | 4.53% |