Hcfyk

You could try LEVITY

https://en.wikipedia.org/wiki/Levity

Levity may refer to

a sense of amusement, the opposite of gravitas

There is nothing wrong with Frivolity as you pointed out. It actually means lack of seriousness. The word Frivolity, from Latin frivolus. It is synonym to levity. It is directly opposite to word serious.

Frivolity (noun): the quality or state of being frivolous.

Merriam-Webster defines frivolous (adj.) as,

https://www.merriam-webster.com/dictionary/frivolous

2.a: lacking in seriousness

His frequent frivolous behavior in the meeting is a big concern.

I generally agree that levity is a pretty direct antonym of gravitas, but "he carried himself with levity" doesn't sound right. So I'll suggest

insouciance
n. Blithe lack of concern; nonchalance.

Flippancy, perhaps?

Flippant - Displaying unbecoming levity in the consideration of serious subjects or in behaviour to persons entitled to respect. (OED)

What about apathy:

1. absence or suppression of passion, emotion, or excitement.




2. lack of interest in or concern for things that others find moving or exciting.

https://www.dictionary.com/browse/apathy

Another option: nonchalance

the state or quality of being nonchalant; cool indifference or lack of concern; casualness.

https://www.dictionary.com/browse/nonchalance

Perhaps 'foppery'? aka: foolish character or action
"The entire exercise was laden with foppery, and as a result lacked any sense of gravitas."

Depending on the actual sentence structure, it could be 'foppish'.
"Nobody will listen to me, I'm far too foppish."

https://www.merriam-webster.com/dictionary/foppery

How about vacuity, silliness, or buffoonery?

This is taken to mean seriousness.

[...]

There's another sense of the word, as the quality of a person having gravitas. In this case, the things they say and do are given significance because of gravitas.

The specific use of "gravitas", this sense of possessing gravitas lending weight or importance to a person leads me to believe you may be extending the meaning of the phrase "air of gravitas" to "gravitas" itself. I therefore believe you are looking for

insignificance, the quality or state of being insignificant, which is itself defined as

not significant: such as

a : lacking meaning or import

b : small in size, quantity, or number

c : not worth considering : unimportant

d : lacking weight, position, or influence : contemptible

(Merriam-Webster)

To put it in context from your question:

What if what someone has to say is automatically disregarded because

of their insignificance?

Consider: mercuriality.

mercuriality

1. the state or quality of having a lively, fickle, volatile, or erratic attitude or character.


2. an instance of such behavior. — mercurial, adj.

The Free Dictionary.

I don't think this can be answered with only one word, honestly.

The media uses gravitas to describe politicians who have a lot of respect or political sway with other politicians even. I wouldn't use any of the words listed above to describe a human being who had the opposite of gravitas. I would call him/her feckless, when the object is a human. https://www.merriam-webster.com/dictionary/feckless

If the object is an object, I agree with frivolous or the other antonyms listed here.

I would suggest Trumpishness. Maybe not common use yet, but it will be. Someone recently said "out-Nixoned Nixon". Everybody knew what that meant.

Carelessness
[kair-lis]
adjective

1) not paying enough attention to what one does

2) not exact, accurate, or thorough:
careless work.

2) done or said heedlessly or negligently; unconsidered:

4) not caring or troubling; having no care or concern; unconcerned (usually followed by of, about, or in)

5) possessed or caused without effort or art; unstudied

6) Archaic . free from anxiety.

Source: Dictionary.com

Triviality

The ODO definition of 'triviality' states:

Lack of seriousness or importance; insignificance.

Here are ODO examples of usage:

‘the mediocrity and triviality of current popular culture’

‘the relentlessness of his triviality is grating’

So in your examples, we might have:

We need to undertake this project without triviality,

He carried himself in a trivial fashion.

What you are getting at is, I think, a kind of weightlessness, and 'ungroundedness' in a person or project, which leads to a desire to avoid or disregard them.