A major asymmetric ice trap in a planet-forming disk IV. Nitric oxide gas and a lack of CN tracing sublimating ices and a C/O ratio $< 1$

[Abridged] Most well-resolved disks observed with ALMA show signs of dust traps. These dust traps set the chemical composition of the planet forming material in these disks, as the dust grains with their icy mantles are trapped at specific radii and could deplete the gas and dust of volatiles at smaller radii. In this work we analyse the first detection of nitric oxide (NO) in a protoplanetary disk. We aim to constrain the nitrogen chemistry and the gas-phase C/O ratio in the highly asymmetric dust trap in the Oph-IRS 48 disk. We use ALMA observations of NO, CN, C$_2$H, and related molecules and model the effect of the dust trap on the physical and chemical structure using the thermochemical code DALI. Furthermore, we explore how ice sublimation contributes to the observed emission lines. NO is only observed at the location of the dust trap but CN and C$_2$H are not detected in the Oph-IRS 48 disk. This results in an CN/NO column density ratio of $< 0.05$ and thus a low C/O ratio at the location of the dust trap. The main gas-phase formation pathways to NO through OH and NH in the fiducial model predict NO emission that is an order of magnitude lower than is observed. The gaseous NO column density can be increased by factors ranging from 2.8 to 10 when the H$_2$O and NH$_3$ gas abundances are significantly boosted by ice sublimation. However, these models are inconsistent with the upper limits on the H$_2$O and OH column densities derived from observations. We propose that the NO emission in the Oph-IRS 48 disk is closely related to the nitrogen containing ices sublimating in the dust trap. The non-detection of CN constrains the C/O ratio both inside and outside the dust trap to be $< 1$ if all nitrogen initially starts as N$_2$ and $\leq 0.6$, consistent with the Solar value, if (part of) the nitrogen initially starts as N or NH$_3$.