This is a surprisingly difficult question to answer... related ones have been asked elsewhere, with few satisfactory answers. Turns out that simple questions like "what is the stellar density in such-and-such a region of space" doesn't often have a very good answer (multiple answers may exist, differing by at least an order of magnitude) let alone more specific things like "how bright is starlight there". I've tried my best, but I haven't been able to get a simple, citeable answer.
Instead of looking at the core itself, I'll look at some other astronomical objects which also exhibit very high stellar densities... globular clusters
There's a relevant article in Astronomy mazagine, which isn't a scientific publication, alas. There are excerpts from the article for free here, including this nice simulation of night on a planet inside the globular cluster 47 Tucanae, as viewed by a regular human eye.
To quote from the quote:
The cluster's suns would combine to give an average sky brightness some 20 times brighter than Earth's night sky at Full Moon
Our sun, as seen from earth, is about 400000 times brighter than the full moon. Clearly, stellar densities just aren't high enough here!
Now, a globular cluster ain't quite a galactic core, but it has some similarities. Near our solar system, the stellar density is about 0.14 stars per cubic parsec. The centremost cubic parsec of 47 Tucanae has about 150000 stars packed into it (though the density drops off sharply... its a tenth of that if you go more than 3 parsecs from the centre). The galactic core by comparison may have a density 100 times higher than that, but even with that it has been suggested that
...there would be a million stars in our sky with apparent brightness greater than Sirius. The total starlight in the night sky would be about 200 times greater than the light of the full moon; you could easily read the newspaper at midnight, relying on starlight alone.
That's a pretty impressive twilight, but one that falls short of your requirements by quite some way. The author does not explain their methodology, so it remains possible that they are incorrect but it seems plausible to a couple of orders of magnitude, and that would still fall a long way short of your requirements!
Lets look at something else that's quite import to your scenario, that you've not really thought about... the particularly hazardous nature of such locations.
When star densities are that high, the chances of you being near to a dangerous sort of neighbour are much higher. 47 Tucanae, for example, may or may not harbour a black hole but it most certainly does harbour 25 pulsars, many of which are millisecond pulsars (for comparison, the nearest pulsar to earth, PSR J0108−1431 is over 420 lightyears away, and it is very weak and spins much more slowly). The creation of such objects is a violent and dangerous event in itself, but they're associated with other unpleasant events such as gamma-ray bursts which are definitely planet-sterilisingly bad things to be near. Binary systems with neutron star companions are known to exist in 47 Tucanae, and that's definitely a forboding combination... certainly, I learned a new term "cataclysmic binary". There are stars which get brighter inexplicably, possibly being some new kind of nova.
The brighter the starlight, the more short-lived giant stars and dangerous supernova remnants there are out there, and the greater the chances of something terminally unpleasant happening to your rogue planet.
Other potentially interesting sources which I didn't manage to back up with more hard data:
This physics.SE answer suggests some sort of bright twilight, though doesn't compare to the moon and I was too lazy to do the maths to compare them myself.
Another physics.SE answer suggests half as bright as moonlight "near" the galactic centre.
This quora answer, suggesting a much brighter sun-like twilight in the galactic centre, though also suggests a more moon-like lights at the edge of the galactic core
You may also wish to look up "Ahad's Constant", which is about how much light from stars other than the sun falls upon earth. The original paper seems to have disappeared, but you might have more luck finding it than me!