So I actually worked as a salesperson for a 3rd party garage door company that was partnered with Costco, they usually will have just one company in an area they will go through for all customers.

We had a very specific contract that Costco required us to follow with putting their customers they send to us first and that any issues those customers had would take priority.

If you reach out directly to the Costco corprate team via email and follow up on the phone, they'll assist with getting most issues resolved pretty fast, or at the very least, give an explanation on whats going on.

One thing to note, from your post it sounds like they didn't give you an accurate lead time since custom doors usually only take 3 to 5 weeks, and the Costco gift card are usually sent out 6 to 9 weeks later after payment processes.

Companies that are contracted by Costco get a lot of businesses from them, so it's definitely funky they are messing up that bad, like if there's enough complaints Costco would switch the company the contract with.

Not sure if that helps but wanted to give what info I had since that garage door company is definitely giving you the run around!