Code change:
I've opened issues for the things that were previously TODO comments in the draft; this removes them from the draft itself.
Issues and for posterity
NAME wrote: Public Keys is maybe worth considering. This too is this is an area for further discussion and consideration should this draft progress.
My take: keep the scope of this narrow. If raw keys are needed - or other certificate types for that matter - then we can define another field.
I concur with Martin's take. The TODO made its way into the individual draft (and subsequently became this issue) based largely on this brief suggestion in one part of an off-list email on the topic, "While I do not know of a use case yet, it might be worth keeping raw public keys in mind as a use case." Which is not unreasonable but I'm going to favor narrow scope here over speculative need.
NAME wrote: might be needed/wanted by the backend application (to independently evaluate the cert chain, for example, although that seems like it would be terribly inefficient) and that any intermediates as well as the root should also be somehow conveyed, which is an area for further discussion should this draft progress. One potential approach suggested by a few folks is to allow some configurability in what is sent along with maybe a prefix token to indicate what's being sent - something like Client-Cert: FULL \ \ \Client-Cert: EE \ as the strawman. Or a perhaps a parameter or other construct of {{?RFC8941}} to indicate what's being sent. It's also been suggested that the end-entity certificate by itself might sometimes be too big (esp. e.g., with some post-quantum signature schemes). Hard to account for it both being too much data and not enough data at the same time. But potentially opening up configuration options to send only specific attribute(s) from the client certificate is a possibility for that. In the author's humble opinion the end-entity certificate by itself strikes a good balance for the vast majority of needs and avoids optionality. But, again, this is an area for further discussion should this draft progress.
I would propose a separate header for the chain that would use the structure List value format.
I'd also argue strongly for enabling the certificate chain (even if as a list of sha256 checksums of entities in the chain). End-entity certificates are effectively namespaced by the rest of the chain, so not including the full chain and/or not tying the end-entity cert to the chain are some of the top foot-guns I've found with people using this.
I like NAME suggestion: separating the EE certificate out is sensible. The assumption here is that the terminating proxy validates the signature from the EE cert when establishing the connection. That makes it different. That said, the origin server might need to assemble and validate the chain, so including all the data presented in TLS is likely necessary. (I don't think hashes are as generically useful; space savings can be achieved with header compression, assuming that you use VERY large tables.)
From a header compression standpoint, I suspect we should note that multiple instances of the header will compress more efficiently than a single instance with a long list, assuming there are multiple possible intermediate cert paths.
NAME wrote: http-core. This should be pretty simple to add, but for the purposes of having a clean PR pulling out the TODOs, I'm moving it to an issue for now anyway.
URL points to draft-ietf-httpbis-semantics for this
NAME wrote: possible with HTTP1.1 and maybe needs to be discussed explicitly here or somewhere in this document? Naively I'd say that the header will be sent with the data of the most recent client cert anytime after renegotiation or post-handshake auth. And only for requests that are fully covered by the cert but that in practice making the determination of where exactly in the application data the cert messages arrived is hard to impossible so it'll be a best effort kind of thing.

Related discussion:
text: 4. <del> TODO: register the "Client-Cert" HTTP header field in the registry defined by http-core. </del> <ins> The "Client-Cert" HTTP header field will be added to the registry defined by http-core. </ins> 5.  References