Ash vs Beige, which is less Brassy?

Neither should be brassy. Ash can have a green, blue or violet base, beige is usually more neutral and not green based.
The names really vary according to the brand, the code is usually more helpful as to the base if it is listed.

I'm looking at these two:
Loreal Excellence Light Ash Brown 6A
Loreal Excellence Light Beige Brown 6BB

To me, the beige brown looks more brassy in the picture.

Ash is more green or green-blue base if below level 8 example 6A

Beige is more violet-blue-rose base example 8V or 8BV or 8BB

but if it is light blonde the ash is more blue based example 10A and beige is more violet based 10V

so the 6A would have more green to cancel out more red
and the 6BB would have more blue-violet and is more neutral

I always prefer Beige over Ash but if you are trying to cancel a lot of red then you may prefer Ash but Ash can be a bit dulling and green

Beige is better too if you have some grey or highlights as Ash could give them a green tinge and Ash does not cover grey well

Ion makes some lovely cool-beige tones like 7V and 9V and they are not brassy at all
they are darker so if you are looking at level 6 then try 7V

also keep in mind that Brassiness often happens when you are lightening too many levels from darker hair
if you are staying within a couple of levels then there is less chance of brassiness