Use REST iControl to upload bundle of key AND cert?

I guess your cert/key bundle is in pkcs12 format? I'd try uploading the bundle (I understood this part is already working) and then use the openssl command line tool on the bigIP to split the bundle into separate key and cert files in pem format. Proper command line for that would be

openssl pkcs12 -clcerts -nokeys -in bundle.p12 -out cert.crt
openssl pkcs12 -nocerts -nodes -in bundle.p12 -out cert.key

Attention: this will produce an unencrypted private key file.

You can try to automate this by executing openssl as an external command through REST, but this might rise some problems with password protected bundles.