I think the issue might be the fixation on having a two minute raid buff window. If the primals were say 25s apiece with 15s for phoenix/bahamut, it'd fit into a 90s cycle which would afford a larger window to each primal rather than them being a flash in the pan. I might be being too conservative here, and perhaps fitting two 40s primals in with a demi would give sufficient room to make engaging. The latter would likely necessitate ignoring a specific primal/cycle, so perhaps the ignored one could be linked to the demi? Naturally this would necessitate the development of a third, but reskinning bahamut again to alex or something could be pretty simple.

Or they could continue alternating bahamut/pheonix, but if you skipped titan maybe it'd upgrade to heritor titan with a different rotation style for the subsequent filler phase. Regular titan could have hard casts and heritor use instants. I think this would allow for a set of rotations, one hardcast focused perhaps with higher damage to compensate, and another more mobility-focused. While I want to suggest the mobility option would be better for prog, I'm pretty sure prog's where you're most likely to encounter enrages so maybe not.

As to what an engaging primal rotation looks like I'm unsure. Maybe every few casts you can build up to casting titan again or have it mesh with the rest of summoner's kit in some way(edit: sorry forgot for a minute) or some aetherflow interaction maybe?