LoL i promise that's not a separate account i'm posting from ct.
but people do seem interested in this (gymnastic/rings makes you feel athletic).
I did straddle front and back lever holds for 2-3 sets as my warmup (including dynamic movements like ice cream makers and inverted hang/lever pullouts). Sometimes I do tucked planche and maltese holds on rings and handstand pushups from wall. Then I do the layers workout. And finish with 25-50 chins/dips on rings.
Basically what you asked for minus the loaded carries.
My personal experience is as you get heavier and if size is the goal, those static holds are a bit to "intense" to be used as warmups. Sometimes I felt activated and would crush the layers workout after doing them. But often they used up quite a bit of CNS "juice" and I wouldn't feel as explosive/violent on the actual barbell portion of workout.
Depending on nutrition, you can also lose the "pump". Doing intense static holds then barbell pressing, for me at least, often resulted in a diminished pump, almost like muscle wasn't responding to the ramp.
Reading your goal as for fat loss, I could see this being not as big a concern. But then your recovery might be hit from any caloric deficit.
Maybe my recovery is off or I can approach this better (CT recommendation?) but here's two cents coming from someone big into gymnastics and lifting..