I just bought some NinjaFlex TPU (Shore 85A) which prints nicely if you print really slowly with it. 10mm/s for 1st layer, 20mm/sec for other.
I tried to do a flow rate calibration and it was horribly under extruded because it was printing the samples way too fast. I had my print settings set correctly for the filament, but the calibration process seems to ignore them.
Is there any way to do a flow rate calibration for such a filament, or is it just not going to work?
Doing calibration for 3rd party filament is no different to 1st party. They use the same tests.
And thus, yes it is possible. Just recently i calibrated my no name 95A TPU to print with a max volumetric speed of 10mm³/s and up to 250mm/s speed for infill etc.
I recommend using Orca Slicer and their extended calibrations
They use the same tests.
And therein lies the problem. 
You mention a no-name 95A at speeds up to 250mm/s. Of course that will work since your TPU obviously supports printing at those speeds.
NinjaFlex does not. The maximum print speed is rated at 35mm/sec, and it’s recommended to go even slower. I have Bambu’s 95AHF and it works great with the calibration. But the calibration print speed doesn’t slow down for the NinjaFlex and it leaves large gaps and spaces, comes out in thin strings, and jams in the print head.
NinjaFlex is 85A which is WAY softer than 95A, and the filament feels like a rubber band. It’s hard to even feed it through the PTE tube. But because of that softness, the extruder just can’t push it through the nozzle at normal speeds, and at the temperature it needs to be printed at. It has to be eased through the nozzle at an excruciatingly slow speed.
But again, the calibration test doesn’t use those slow speeds. At least not in Bambu Studio. Does Orca slicer allow slowing the speed way down for the calibration prints?