My take is thus:

The flange rating charts in API 6A, 6AF1, 6AF2, etc. are constructed from the convenience of SES (the group who did the work) from a standpoint of FEA and experiment.

It's not constructed from the convenience point of a stress analyst.

My recommendation is thus:

1. Place a CNODE anchor at every flange pair to check.
2. Place a node name at these points that include the size (if you have multiple sizes) and api pressure class (if you have multiple pressure classes).
3. Export from CAESAR a local restraint summary, isolating your flanges.
4. For each size + pressure class, find the maximum Fx from the local restraint summary.
5. This Fx is the axial load. Note: Careful construction of your CAESAR model should allow you to discern whether this is compression or tension, but I tend to assume compression = tension for conservatism.
6. Round up to the next tension level that's listed on the API 6A chart.
7. At your design pressure rating, find the correlating bending moment.
8. This tension and bending moment combination will serve as a "recommended" combined limit.
9. Calculate sqrt(My**2+Mz**2) for all flanges to be checked. This is your bending moment.

Notes:
1. If any flange exceeds bending moment or tension limits on the chart outright, obviously you will need to reroute to fix this.
2. The "recommended" combined limit is a soft limit. If any flanges have higher bending moments but much lower tensions, then check these outliers manually against the chart.