The tooltip displayed above bundles while selecting an item reflects neither the data set by the item's tooltip_style component nor the bundle's tooltip_style component.
Steps to Reproduce:
Run /give @s brush[minecraft:tooltip_style='example']
Put the Brush into a Bundle
Observe the tooltip of the item in the bundle
Alternatively:
Run /give @s bundle[minecraft:tooltip_style='example']
Put an item into the Bundle
Observe the tooltip of the item in the bundle
Can confirm. Also applies to custom textures with resource packs:
[media]