Does copper(III) nitrate exist?
Copper(III) nitrate cannot be obtained from aqueous nitric acid, and likely doesn't exist.
Reaction carried under strongly oxidative conditions between $\ce{Cu(NO3)2}$ and fuming $\ce{HNO3}$ yields in nitrosyl copper(II) trinitrate $\ce{[NO+][Cu(NO3)3−]},$ [1, 2] sometimes written as adduct $\ce{Cu(NO3)2 · N2O4},$ which is contradictory to the crystal structure consisting of copper(II) centers octahedrally coordinated by nitrate groups and the voids in cross-linking mesh of octahedra are occupied by nitrosyl ions [3]:

Figure 1. Crystal structure of nitrosyl copper(II) trinitrate $\ce{[NO+][Cu(NO3)3−]}$ (ICSD-422033). Color code: $\color{#3050F8}{\Large\bullet}~\ce{N}$; $\color{#FF0D0D}{\Large\bullet}~\ce{O}$; $\color{#C88033}{\Large\bullet}~\ce{Cu}$.
According to Wiberg's Anorganische Chemie [4, p. 1707], oxidation of $\ce{Cu^2+}$ to $\ce{Cu^3+}$ is possible only with strong oxidation agents $(E^\circ(\ce{Cu^2+/CuO+}) = \pu{+1.8 V}),$ and copper(III) is highly unstable in water and has a tendency to oxidize the medium $(E^\circ(\ce{H2O/O2}) = \pu{+1.23 V}).$
The reduction potential or $\ce{Cu^3+}$ can, however, be reduced down to $\pu{+0.45 V}$ by coordinating it with appropriate strongly donating ligands (which nitrate really isn't).
Examples of such complexes are:

$\ce{K3CuF6}$ (oxidant: fluorine $\ce{F2}$; $\ce{CuF3}$ is unknown);
$\ce{K5[Cu(IO6H)2]},$ $\ce{K5[Cu(TeO6H2)2]},$ $\ce{Na5[Cu(TeO6H2)2]}$ (oxidant: peroxodisulfate $\ce{S2O8^2-}$ in strongly alkaline medium);
$\ce{Na[Cu(OH)4]},$ $\ce{KCuO2},$ $\ce{Ba(CuO2)2}$ (oxidant: hypobromite $\ce{BrO-}$/oxygen $\ce{O2}$; $\ce{Cu2O3}$ is unknown).
Cu(III) complex of 2-ethoxy-5,10,15,20-tetrapentafluorophenyl-3,7-diaza-21,22-dicarbaporphyrin (Cu(III)-$\ce{N2CP}$, a doubly N-confused porphyrin) [5]
A variety of Cu(III) peptide complexes [6, pp. 213–223]

References

Znamenkov, K. O.; Morozov, I. V.; Troyanov, S. I. Synthesis and Crystal Structure of Copper(II) Nitrato Complexes $\ce{NO[Cu(NO3)3]},$ $\ce{Na2[Cu(NO3)4]},$ and $\ce{Ag2[Cu(NO3)4]}.$ Russian Journal of Inorganic Chemistry 2004, 49 (2), 172–179. (in Russian)
Gagelmann, S.; Rieß, K.; Wickleder, M. S. Metal Oxidation with $\ce{N2O5}$: The Nitrosylium Nitrates $\ce{(NO)Cu(NO3)3},$ $\ce{(NO)2[Zn­(NO3)4]}$ and $\ce{(NO)6[Ni4(NO3)12](NO3)2·(HNO3)}.$ Eur. J. Inorg. Chem. 2011, 2011 (33), 5160–5166. https://doi.org/10.1002/ejic.201100565.
Blackwell, L. J.; King, T. J.; Morris, A. Crystal Structure of the Copper Nitrate–Dinitrogen Tetroxide Adduct $\ce{Cu(NO3)2 · N2O4}.$ J. Chem. Soc., Chem. Commun. 1973, No. 17, 644–644. https://doi.org/10.1039/C39730000644.
Wiberg, E.; Wiberg, N.; Holleman, A. F. Anorganische Chemie. Nebengruppenelemente, Lanthanoide, Actinoide, Transactinoide, 103. Auflage.; De Gruyter: Berlin; Boston, 2017; Vol. 2. ISBN 978-3-11-049573-7. (in German)
Furuta, H.; Maeda, H.; Osuka, A. Doubly N-Confused Porphyrin:  A New Complexing Agent Capable of Stabilizing Higher Oxidation States. J. Am. Chem. Soc. 2000, 122 (5), 803–807. https://doi.org/10.1021/ja992679g.
Karlin, K. D.; Tyeklár, Z. Bioinorganic Chemistry of Copper; Springer Netherlands: Dordrecht, 1993.