A faster rate for the first few days might help then, that way you can quickly build a small collection. maybe at first it's 100% to get one item and 70% to get a second. if you roll them at the same time, you could also make sure they roll for different parts, that way there's some guarantee of a spread. you could reduce the odds on getting the second part to 0, then reduce the odds on rolling one. By gut I'd only reduce when items is obtained. rate of reduction would depend on total number of items
you could also weight the odds of the parts based on the power, and boost the rewards based on performance in the judging, i.e. first place increases the odds on rarer parts or increases the likelihood of additional parts. But those are ideas for later, or maybe already in Breeding Pool.
I know I got a bit unlucky here, and I'm not worried if there are only 50ish parts for a game jam game. That's enough to get the idea across.