Here are some good ideas for you:

This blog shows Pale Oak in lots of scenarios, and recommends colors that work well with it. They do comment that this color does not work as well in a monochrome color scheme. https://heatherednest.com/pale-oak-benjamin-moore/

There's a nice color palette here: https://themorrismansionblog.com/pale-oak-benjamin-moore/#Coordinating_Colors_for_Pale_Oak

Per this blog you are correct to avoid green hued colors. The Mount St Anne looks lovely with the BM pale oak. https://thecolorconcierge.com/benjamin-moore-pale-oak/

Sooo many options.

My first question is what colors do you like? What colors are you using?

Pale oak can go with Blues, Greens, Marrons, Browns,....

As long as the accent color is a bit grayed down it appears lots of colors will work with Pale Oak.

Where is the Pale Oak and what are your plans for the darker accent color?

@Boxerpal I have navy blue sofa in living room. I am thinking to do coffered celing with accent color of pale oak. So may be a darker shade of pale oak in same family.

Have you considered the reverse? The darker shade on the wall and the pale oak on ceiling?

This room is not thunder on the walls, using it to give you an idea of going darker below and lighter above.

Ask your ben moore painter to add 25% more black to the pale oak to give you a slightly darker color.

Are you keeping the light fixture?

@elcieg pale oak on walls and accent color on squares in the ceilling. Pale oak is not painted yet.

@eclieg Thunder looks great. Will check it out! Thanks.

@Boxerpal That is a good idea! Will check that out. Yes, light fixture stays. Thanks.

BM collingwood may be used as accent. Will try.

IMO nothing you have selected so far works with your existing paint colors. It's apples and oranges. You'd really have to repaint everything to move all the warm tones currently in your home over to the very cool tones you have selected for your seating and area rug.

The room was lovely to begin with. You could have selected contemporary items that worked with the existing paint colors, but this is not it.

Contemporary in a more neutral palette would have worked better in this room.

Just to help a bit with color. I am a Certified Color Specialist and here are the comparisons of the Color notations (DNA) of BM Pale Oak and S.W. Alabaster. You can see the LRVs are quite a bit different. Alabaster being more reflective. You can also view the other color notations and we can discuss further if you are interested.