To the point of trying out a different lens...
Here is a series of shots taken with a Tamron 90mm f/2.5 macro lens at its closest focus distance (1.3') to record a 1:2 repro ratio image on a 5D camera. Subject area is white bond paper illuminated under CFL known by me to have a color temp of 2700K. Shots were recorded in RAW and imported into Lightroom.
Two points I'd like to raise to your attention:
- Shots 1 & 3 are portrayed 'as shot', with #1 at f/2.5 and #3 at f/11. One can see that at f/2.5 there is noticeable non-uniformity of field, whereas at f/11 the field looks quite uniform. So brightness of center is NOT inherent to macro lenses.
- Shots 2 & 4 are Lightroom virtual copies of #1 and #3, respectively. But exposure has been altered (in postprocessing) by +3EV, to make the white bond LOOK WHITE. One of my pet peeves on POTN are shots taken by amateurs of white objects, underexposed so that the 'white' is visibly gray. That is not 'good photography' to underexpose an image, especially when it is pointed out that underexposure leads to inferior signal:noise characteristics in the shots and so many folks have a beef already about Canon inferiority in s/n vs. Sony and changing brands for that reason. PLEASE...Let's not portray the world darker than it is, and make our cameras look noisier than they need to be.
Your meter tries to take what it sees as a target, and suggests an exposure which CAPTURES IT GREY, so use EC to tell the meter your subject area is NOT grey, and to give more exposure to the scene so that white records as white. [Stepping down from soapbox]

You shot on APS-C with about 15mm x 22.5mm sensor, which should mask any lens non-uniformity of field brightness because it is not using anything but the center 60% of its image circle. If anything, my FF image should have highlighted non-uniformity because it captures more of the lens' image circle in its 24mm x 36mm sensor!
If we adjust your photo (#1), increasing exposure by 2.5EV resulted in this result (#2), in which field non-uniformity is masked