It is difficult to provide the exact solution without looking at the logs. But going through the comprehensive details you provided following are few things you could consider to execute
a) Remove the second NIC and try whether you are able to access the iSCSI target using the NICs one at a time.
b) That is first connect only the Intel NIC as the uplink for the VMKernel port used for iSCSI access and checke for the iSCSI connectivity.
c) Repeat the test by removing the Intel NIC and adding the Broadcom NIC
d) If you are able to access iSCSI target in both the case 'b' & 'c', check whether you are trying to mix-up an iSCI offload adpater with a regular NIC card in your configuration?
e) In case you are NOT able to access iSCSI target in any of the case 'b' & 'c', you have identified the point to troubleshoot and you need to fix the network connectivity for the specific NIC card
Also check the following
a) End to End Switch connectivity, if the NICs are connected to different switches
b) VLAN tagging if it is applicable
c) Jumbo frame configuration
~Sudhish