Adding certificate chain to p12(pfx) certificate

Own answer.

I figured out how to do this with OpenSSL:

openssl pkcs12 -in certificate.p12 -out clientcert.pem -nodes -clcerts
openssl x509 -in trusted_ca.cer -inform DER -out trusted_ca.pem
openssl x509 -in root_ca.cer -inform DER -out root_ca.pem
cat clientcert.pem trusted_ca.pem root_ca.pem >> clientcertchain.pem
openssl pkcs12 -export -in clientcertchain.pem -out clientcertchain.pfx