It's copyright infringement, and that's what's called a "strict liability" offense, meaning how little you take and whether your intent was relatively innocent is irrelevant.
Since you're only using this for a dozen or so invitations to your kid's party, the likelihood of you getting caught and hit with a $30,000 or even $150,000 statutory damage award for willful infringing is close to 0. but it's not 0. And I don't typically give parenting advice, but shouldn't you try to set a good example for your kid, and not instead show them that stealing's ok if you only steal a little? You've probably got a choice of buying licensed invitations with this cartoon on it, so consider doing that.
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Per 17 USC 106, copyright protection includes 6 enumerated rights. Your printed invites would infringe the first three:
"1) to reproduce the copyrighted work in copies or phonorecords;
(2) to prepare derivative works based upon the copyrighted work;
(3) to distribute copies or phonorecords of the copyrighted work to the public by sale or other transfer of ownership, or by rental, lease, or lending; "
The fact that you are not selling these is not a defense, as you can see from the statutory language just quoted.
It is the commercial effect upon the copyright owner that matters not your profit or lack of profit. If the cartoons are protected by copyright REGISTRATION, the owner would be entitled to statutory damages of $750-$150,000 per item. While you would likely fall at the low end of that, still with a large number of invites you could be talking serious money.
Make your own artwork or buy licensed artwork. Don't steal someone else's intellectual property such as their copyright protected cartoons. Proceed at your risk if you do. You may not be noticed by the copyright owner or the owner may have bigger infringements to address, but that does not make you legal, rather only lucky. Do you feel lucky?
I am not your lawyer and you are not my client. Free advice here is without recourse and any reliance thereupon is at your sole risk. This is done without compensation as a free public service. I am licensed in IL, MO, TX and I am a Reg. Pat. Atty. so advice in any other jurisdiction is strictly general advice and should be confirmed with an attorney licensed in that jurisdiction.