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My mother just blew me off.

My mother just blew off bringing the car seat.

You done been blown off.

The verb phrase blow off can take an object immediately after blow (especially for pronouns or names) or after off (for noun phrases denoting tasks like work). According to the Oxford English Dictionary (in "blow, v.1") it means both disregarding someone else and shirking a job or duty:

transitive. to blow off:

(a) (now U.S. slang) to rebuff, to reject the advances of (a person); to ignore, disregard, dismiss;

(b) U.S. slang, to shirk or evade (a job or duty), to stay away from (school or work) without permission or good reason.

If you're British, you may have recourse to blow out, with similar valences:

4. transitive. Chiefly British.

To let (a person) down or shut (a person) out; to rebuff, reject the advances of (a person). Also: to fail to keep (an appointment), esp. deliberately, to jilt; to shirk. Cf. earlier to blow off at Additions.

Blow off especially pertains to your situation because your mother has both disregarded your concerns and has shirked the task at hand. Blow out might pertain more if she missed the appointment entirely, rather than neglecting some part of meeting you.

Thanks for offering some suggestion as to how you might want to use the word/phrase that you seek. Deriving the meaning from your detailed story and example sentence, I think your mother just "disregarded" you.
I have also attached the freethesaurus.com link using the same word. It offers several other clever and nuanced suggestions.

There is one additional idiom that seems to fit your latest edits and examples. In every situation they seem to be "poo-pooing" your arguments/requests. It means to outright dismiss someone/something without any due consideration. This fits your examples now very well. Please consider it.

My mother just disregarded me.

When I reminded her about the car seat, she just "poo-pooed" me.

The parent who brought up the no-nut thing was "poo-pooed" by
some of the others.

https://www.thefreedictionary.com/disregard

https://www.freethesaurus.com/disregard

https://idioms.thefreedictionary.com/poo-poo

Guff[guhff]
noun Informal.

1) empty or foolish talk; nonsense.

2) insolent talk.

Source: Dictionary.com

———

Grief
noun

3) informal trouble or annoyance:

people were giving me grief for leaving ten minutes early

Source: Dictionary.com

———

The phrasing I would suggest would be:

When I reminded her about the car seat, she just gave me guff (grief).

The parent who brought up the no-nut thing was given grief (guff) by some of the others.

In my interpretation:

• guff implies the person doesn’t know what they are talking about.

• grief implies the person is affecting you.

Ignore, perhaps?

My mother just ignored me.

Or other phrases with similar meaning -

pay no attention

take no notice

brush aside (my) concerns

It's a bit like intimidation - you have a concern you think is legitimate, but they're trying to bully or browbeat you into submission.

Flout?
ARCHAIC
mock; scoff.
"the women pointed and flouted at her"

I quite like 'dismissed my concerns', though it doesn't roll off the tongue. I think 'diss' derives from 'disrespect'

The problem is I don't think there's an English word for exactly this, because you said your mother thought you were being 'fussy' about this. It certainly was not such strong opposition to suggest 'flak' which would be far too strong.

Flak is like military shells (figuratively). Saying you were 'dissed' doesn't sound too 80sish to me, just not quite right. That's more like she put you down over it (not what happened). I think the closest you can get to it is something a lot milder, like 'she gave me a little heat about it'. But I wouldn’t say 'heat' alone, just 'a little heat'. It needs to be a word or phrase that suggests you were worrying unnecessarily about something, in her opinion.

Consider dismiss.

dismiss
2 Treat as unworthy of serious consideration.
‘When I campaigned for Police Community Support Officers in 2000, as the Mayoral candidate for London, the idea was dismissed by my opponents.’
- ODO

In the context of the examples you raise, dismiss tends to apply more naturally to the ideas rather than the people. Dismissing the person is a lot more offensive than dismissing the idea, though in the second example below, it could be read as a metonymic dismissal of the idea.

• When I reminded her about the car seat, she just dismissed [it / the idea].


• The parent who brought up the no-nut thing was dismissed by some of the others.

To dismiss an idea is to consider it so trivial that it doesn't bear further discussion. You can say that someone's attitude was dismissive, or even that they were dismissive. To complain that you were dismissed often implies that you were offended by the dismissal.

When using dismiss to mean the setting aside of something trivial, it's usually pretty clear that the one doing the dismissing considers the thing to be trivial. Whether it actually was or wasn't trivial relies on the context.