Context without cancel propagation

Can anyone advice how this is supposed to be done?

Yes. Don't do it.

If you need a different context, e.g. for your asynchronous background task then create a new context. Your incoming context and the one of your background task are unrelated and thus you must not try to reuse the incoming one.

If the unrelated new context needs some data from the original: Copy what you need and add what's new.

Since context.Context is an interface, you can simply create your own implementation that is never canceled:

import (
    "context"
    "time"
)

type noCancel struct {
    ctx context.Context
}

func (c noCancel) Deadline() (time.Time, bool)       { return time.Time{}, false }
func (c noCancel) Done() <-chan struct{}             { return nil }
func (c noCancel) Err() error                        { return nil }
func (c noCancel) Value(key interface{}) interface{} { return c.ctx.Value(key) }

// WithoutCancel returns a context that is never canceled.
func WithoutCancel(ctx context.Context) context.Context {
    return noCancel{ctx: ctx}
}