Rotation energy barrier for carboxypyrazolate
One of way to answer this question is to put it in some QM software. The system is small enough and has higher symmetry so computations shouldn't take to long, even on an older computer.

For co-planar geometry the energy calculated at B3LYP/6-31G(d,p) is –492.132206748 a.u.
The geometry with dihedral angle of 90° between $\ce{CO2}$ and $\ce{C3N2}$ is –492.116719625 a.u. That makes roughly 41 kJ/mol (10 kcal/mol).

Now if you really want to be sure if that's the right energy you should make frequency calculations on both structures to find out if they are minima or transition states. This can take a bit longer (I haven't done it) and I would expect that the better structure (with lower energy) should have no imaginary frequencies and the higher one should have one (TS). If by some accident both of them are minima than TS has probably a dihedral angel somewhere around 45°.

Of course this is only computation. It gives as clues but not the exact answer. You can go for higher level of theory, add solvent modelling and... or you can take your compound and try to make temperature depending NMR spectra. With a bit of luck you can freeze your TS and find out how high is your barrier.

p.s.

It would be good if you mention in your post what tools (in this case computational) are available to you. I really would prefer to explain how to get the number instead of giving just dry answer.