Unlike onchain transactions Lightning nodes do not get a global view of all transactions. They only see transactions that they send, they receive and they route across the network for others. When they help route a payment they don't even see the final destination of the payment, merely the next node along the route.
I'd like all payment transactions sent to a given recipient via the LN protocol to be publicly indexed and independently verifiable.
You would only be able to index a recipient's transactions if that recipient published this information. There is no obligation to do so and no alternative.
Is it possible to get this information from the Lightning Network protocol? Is it already stored/indexed on the sender/recipient's node (so the sender and/or the recipient could relay to a node for inclusion)?
Not from the normal running of the protocol. As I said you would need the sender and recipient to make this information public.
What LN transaction information is usable to provide a verifiable public index of received payments? I am thinking the invoice and related pre-image "receipt" could work. Or am I barking up the wrong tree?
The network topology and channel capacities are public and gossiped across the network. But the channel balances and payments that are routed across the network are not public. The receiver generates the invoice and the sender constructs the route so this data could theoretically be made public. The hash preimage that is revealed across the route from the receiver back to the sender could also theoretically be made public. However, in practice this would be a significant privacy leak and it would be difficult to convince Lightning nodes to do this without any obvious gain for them.