You could "rent" space by setting up by-the-hour cloud services, but you'd probably just trade a compute problem for a bandwidth problem. You might want to make sure you know where the bottleneck is before trying to add resources - if it's your LAN you can save time by moving closer to the files and adding a faster connection. But most likely it is CPU or memory.

Can you borrow some servers from a local computer store for a few weeks? It's not outside the realm of possibility. Call them all, they can only say "no". Or maybe some businesses in the area might have some spare resources. I'd say leverage your status as a pro bono organization for some goodwill help.