Are you modeling drag? If you are not, you don't even need to model the Earth's rotation because the J2 effect depends on latitude only. This is a low fidelity simulation (there are lots of effects other than J2; e.g., drag, third body effects, higher order gravity terms, solid body and ocean tides, solar radiation pressure, relativistic gravity, ...)
But if you are modeling drag, you'll need the satellite's geodetic location (geodetic latitude, longitude, altitude) and the local apparent solar time at that latitude and longitude to compute the atmospheric density at the satellite's altitude. That raises the ante on the infrastructure you need by quite a bit. At a minimum, you'll need a semi-realistic model of the Earth's rotation, both for the ECEF information (latitude and longitude) and for time.
And you'll need a model of time. Time measured according to how the Earth rotates and time measured according to the ticks of an atomic clock are two different things. You'll need to model this, at least to some extent. If you are using physics-based equations of motion, time in your simulation should be in sync with time according to that atomic clock. Effects from the Earth (non-spherical gravity, drag, ...) and where the satellite is with respect to the rotating Earth should be in sync with time according to the Earth's rotation.
An easy, low fidelity model of time: UT1 is within 0.9 seconds of UTC; for a low fidelity model you can ignore that (i.e., assume UT1=UTC). UTC is currently 37 seconds behind TAI, which in turn is 32.184 seconds behind Terrestrial Time (i.e., assume UTC=TT-69.184 seconds). Terrestrial Dynamic Time deviates from TT by a tiny fixed offset (~7e-5 seconds, which you can ignore) and by a couple of sinusoids whose magnitude is in the millisecond range (e.g., assume TDB=TT).
An easy, low fidelity model of Earth orientation: This is a bit tougher, especially since the Standards Of Fundamental Astronomy (SOFA) code makes it so easy to use a high fidelity model, often much higher fidelity than you need. (The computation of precession and nutation is easy to code but it is not cheap computationally.)
What follows is complete heresy: (1) Use the above simple model of time to compute UT1 and TT, (and also TDB if you want third body effects). and (2) replace the polar motion terms in the SOFA models with zero. (One term has already been zeroed out, ΔUT1, by assuming UT1=UTC). With ten, maybe twenty, lines of code, plus the SOFA library, you have just bumped your rather low fidelity model to moderate fidelity. If you want third body effects, use C-SPICE to compute the location of the Moon and the Sun with respect to the Earth. C-SPICE uses TDB as its time base, but TT will suffice for a moderate (not low) fidelity simulation. Note very well: For third body gravitation, you want C-SPICE to not compute aberration effects.
Aside #1: At 400-600 km altitude, you really do need to model drag if you want to have any hope at realism.
Aside #2: At some point in time (and four to six months for a satellite at 400-600 km altitude is well beyond "some point"), it doesn't matter how high fidelity one makes ones simulation. There is no hope for a realistic projection of a 400-600 km altitude satellite's ECEF position six months into the future. You might get the altitude right. Latitude & longitude: Not really. One burp from the Sun can make the Earth's upper atmosphere increase in density by orders of magnitude, and those burps are unpredictable. That said, those solar burps are less likely for the next few years given the Sun's currently weird quiescent state.
