Smoothing out texture light reflections from fine art photos

In that case the "cheating" option may be the best bet.

Put the camera on a steady tripod. Set up the lights as before, make sure they are pointed to the far edge. Turn on one of the lights. Shoot. Turn it off, and turn on the other. Shoot.

Stack them in Photoshop, zoom in and nudge to make sure they are perfectly aligned. Then put the top layer in Darken blend mode.

I assume you shoot raw for this, not jpeg. You get the best result if you compensate for the light falloff with a simple gradient in ACR/Lightroom, and otherwise match the two exposures closely.

Just an additional comment to clarify: the main problem you have here is one of angles. You have a too steep angle from the lights to the artwork, hence the hot spots. This can happen if it's a very large work - or a small room - and you don't have enough space to get the lights out to the sides. Very often you have more room vertically, and bouncing the light from the ceiling solves the problem.

Here's a neat trick: put a steep curves layer on top. This will immediately reveal uneven lighting and other problems. Add anothe layer underneath to correct using masks and soft brushes, painting in the correction.

What you have here is classic "hot spots". I don't know how big this is, or how much of the wall it takes up - but in cases like this I always try to bounce the light from the ceiling. If that's not possible, position your lights as far away as you can to the sides. Diffusion (soft boxes etc) are actually not recommended here - it will only increase unwanted reflections. What you want here is point light sources. The smaller the better.

If everything else fails, cheat. Shoot twice, with light from either side. Stack them in Darken blend mode. You need to have them both corrected first for light falloff.

You mention polarizers, but have you tried cross-polarizing? In addition to the polarizer on the lens, you put polarizing film on the lights. Obviously, that needs to be flash units, or perhaps LED; otherwise the heat will melt the film. Mounted at right angles to each other, this will kill almost any reflection. You also lose 4-5 stops of exposure. Here's an example:

Photographing artwork is a huge subject. I work as photographer at an art museum, and I constantly need to invent new techniques. There's always a new problem. But yes, the eternal enemy is reflections. 80% of it is getting rid of reflections - the rest is getting rid of shadows. Or the other way round - it depends 😉

Can't you scan them?

Since you do not have problem to photograph again paintings I will recommend to you to try with diffusers which can hep you to photograph without reflections. It is not expensive gear and you can also DIY. Search on internet on topic and probably you will find solution. Personally I am using paus or tracing paper from time to time to soften source light and avoid highlights.