Running a relay to help could be a good start. It's pretty simple[0][1]. The risks associated with being a Tor exit node do not apply to all relay types and the network can be helped greatly by running a standard relay too (standard relays only pass encrypted traffic from A to B).

[0] https://www.torproject.org/docs/tor-doc-relay.html.en

[1] https://www.torproject.org/docs/tor-relay-debian.html.en

If I were to run an exit node or a relay node, how would I mitigate the rare-but-potentially-life-destroying risk that my machine is identified as trafficking in illegal bits, due to relaying/exiting traffic of that nature from the anonymized network? I would be happy to donate bandwidth, but that risk is something which I don't feel is acceptable.

A normal, non-exit relay node will not be identified as "trafficking in illegal bits" by any sane police force. A normal relay will trade from A to B in only encrypted traffic and will not touch websites on the behalf of users, only other relays.

It is rare that an exit node is raided but there has been no known occurrence in which a normal relay was raided.. nor would it make any sense for there to be. Many people run non-exit relays as they are uncomfortable with the risk of exiting for the Tor network, perhaps you should be one of them.

That isn't a risk for relay nodes, because you would be relaying encrypted traffic. It would just be noise.

For exit nodes, the issue of legal liability for routing other's traffic is less clear. You'd basically run the same risks as providing an open wireless network: https://openwireless.org/myths-legal

You can run exit nodes anonymously. Prepaid, anonymous credit cards are available everywhere now.