Should I List My One Product on Amazon or Not?

I manufacture one product and as of now it is only available to purchase on my website. I am considering listing it on Amazon too (nowhere else) — and I just went through the process of setting up a Seller Account and Brand Management.

The problem is… I have to set a higher price on Amazon to cover their fees, and I just read that Amazon does not like when Sellers sell for cheaper elsewhere, and that they penalize Sellers for this by making it harder for customers to buy from you. However, all the stories I read involved products that multiple sellers offered, which is not the case for me.

QUESTION: If I undersell the Amazon price on my website, will Amazon still remove the BUY Button on my Amazon listing even if I’m the only seller on Amazon offering the product? Or worse — will Amazon remove or hide my listing, or make it impossible to find?

Honestly, I just want to list the product on Amazon for the exposure; I don’t expect anyone to buy it at that higher price. But if Amazon is going to make it so no one can buy my product, or even see my listing, then why should I pay them $468 a year ($39/month) to list my product there?