SSL Handshake failed | client certificate authentication

The server is rejecting the connection right after the client sends its certificate, which implies that the server is rejecting the client's certificate. The most likely cause here, given that browsers work, is that your Staging_NPS_Root does not contain the full PKI trust chain. Browsers will usually work if they have all of the CAs in the chain because they'll send these CAs (except for the self-signed root) in the handshake. Your web services may not be doing this. You can tell for sure by doing an ssldump capture on the client side traffic:

ssldump -AdNn -i [client side VLAN] port 443

In browser traffic you'll see the client send the client cert and at least one subordinate CA cert. To fix then, you can either add the missing subordinates to the Trusted Certificate Authorities bundle file on the F5, or to the web services clients (if they support sending the subCAs).