I've MPLS LSP between Cisco 1 and Cisco 2 and between Cisco-3rd part device (Cisco 2 is having Multi-instance ISIS)
cisco1 - Cisco 2 has a iBGP-Labeled unicast peering
Cisco2 and 3rd party device also has iBGP LU peering
There is a BGP VPNv4 peering between Cisco1 and 3rd party device
VRF 'RED' exist on Cisco1 and Cisco 2
I see that on the Cisco 1, the routes advertised by 3rd party device are installed in the VRF 'RED' routing table but in the Cisc1' CEF table I see only the VPN label for the routes advertised by 3rd party device
I believe the CEF table should also have the IGP label (LDP/SR label to reach the 3rd party device advertised by Cisco 2)
The "unresolved" in the output you provided tells me that you do not have a static route for the /32 BGP LU directly connected next-hop from cisco1 to cisco2. This is required on XR as specified in the following document.
For Cisco IOS-XR over Inter-AS link, there is a different logic as compared to that of Cisco IOS. It is required to configure a static /32 route to ASBR1's interface, so that MPLS label is bound for a /32 prefix. If this is not done then the control plane will come up, but the traffic will not be forwarded.
Thanks for the additional information. Since you use BGP LU as a replacement for LDP/IGP, the session needs to be establish the BGP LU session using the directly connected peer (126.96.36.199) rather than on the loopback address. You also need to enable mpls on the all interfaces using the "mpls activate" command under BGP.
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