Grease pencil isn't color managed in the viewport, so no, it's a pitty but there isn't a solution for that.
Unless this question about color management will attract more people and devs to really dive into it.
I've tried to push this specific question about grease pencil, but see by yourself how I didn't even had one response:
Perhaps it's not well formulated, and my knowledge is limited related to grease pencil, coding... but still the question remains.
To be a bit more specific related to your question, what you see in your 1st picture is, on the left part,
a well color managed mesh and a grease pencil stroke not color managed at all. a mesh color managed with filmic and a GP stoke color managed in default (clamped sRGB)
Resulting in the right, in a render that not correspond to what you see because everything is color managed with Filmic.
In the second image, the solution to set color management to default (what a poor choice to name a clamped sRGB color space), isn't a solution at all, since light will behave incorectly.
You can choose to change all your grease pencil color values by guessing and hope, sorry, pray that one day all part of Blender will be correctly color managed (I don't speek only about GP).
Sorry, even with my little knowledge about color management and spaces, I still make mistakes, it's not that GP isn't color managed, but it's locked to defaut (clamped sRGB)