tls – Should the client send the CA chain with the client certificate after ServerHello?

You are responsible for sending enough string for the server to connect your certificate to a trusted root.

For TLS 1.2, this is described in RFC 5246. The client certificates are defined in 7.4.6, which states in part:

Client certificates are sent using the defined certificate structure
in section 7.4.2.

And if you look at 7.4.2, he describes the structure of the certificate as including the liste_certificat (it's me who points out):

It's a sequence (chain) of certificates. The certificate of the sender
MUST come first in the list. Each subsequent certificate MUST directly
certify the one who precedes it.
Because certificate validation requires
root keys are distributed independently, the self-signed system
certificate that specifies the root certification authority MAY be
omitted from the chain, assuming that the far end must
already owns it
to validate it in all cases.

In short, the server is supposed to have the root approved, but it is not required nor should it have intermediate certificates that may be required. The customer is required to provide them if he wishes the verification to proceed reliably and smoothly. (And the same goes for certificates that the server sends to the client).