When you are chitting seeds, or even growing seeds early on after chitting, the seeds are not relying much on whether the media is peat, non-peat compost, or something else. Have you considered just sowing seed in a good clean sand to start them germinating? That solves the problem of peat-free vs. not peat-free compost, if that concerns you.
Many farms in northern Indiana are extremely sandy (geologic deposits of Great Lakes beach sand and dunes), but farmers do grow crops on such soil by regularly feeding dilute fertilizer with irrigation. It is almost like hydroponics.
On the scale of your seed-sprouting bench, you could easily do this, just provide a little dilute water-soluble fertilizer when you water. Pot on to a larger container with regular compost after the seedlings have a good start.
I have been thinking of doing this next year, in part due to the lower cost of sand vs. seed-starting mix.