Are gravitational waves part of dark energy? Do the current models of the universe consider the energy release of gravitational waves?

E.g. iirc, the last black hole merger detected had a gravitational wave energy equal to ~3 solar masses (hope I'm not mistaken in this fact).

Same for other mergers, and supernovae: the energy related to their gravitational waves should sum up to something.
Gravitational waves cannot be part of dark energy, because -- just like electromagnetic waves -- their energy density dilutes as the universe expands: both due to the simple expansion of space and due to the cosmological redshifting that the expansion produces.

Dark energy, on the other hand, has an energy density that remains constant as the universe expands (truly constant if it's the cosmological constant, very close to constant if it's something stranger like quintessence), so it can't be made up of GWs.

GWs probably don't contribute much to the energy density we attribute to dark matter, either, since dark matter is diluted by the cosmological expansion but not by redshift. So if the energy density attributed to DM ten billion years ago were due to GWs, then it would be significantly lower than the DM energy density we measure today. (Also, current cosmological observations indicate significant dark matter was present very early in the universe, before the star formation that was necessary to create binary black holes in the first place.)

The GW energy from a binary-black-hole merger is enormous, yes; but those are very rare events. The current estimate from LIGO observations is about 50 or 100 such mergers per year per cubic gigaparsec, which translates into a rate of less than one per million years per galaxy. This paper by the LIGO and Virgo Collaborations has estimates of the current GW energy density in units of the current energy density of the universe (their Figure 1): 



The green curve shows the estimated energy density from binary-black-hole mergers as a function of the frequency of the GWs; the red curve shows the contribution from binary-neutron-star mergers. The combined peak is always less than $10^{-8}$ of the critical energy density, so even summing up over frequencies, we have an energy density much smaller than that due to ordinary matter ($\Omega \sim 5$%), so it's probably safe to ignore its effects.