I was wondering if they look at your old PTA’s and cross check the information.
Only HPD can answer that.
There's no difference if you wait for a call to bring in the letter vs. volunteering it. If it's been 10 years since you needed to bring it in, it's plausible that you didn't remember it 10 years later.
If you were not denied a permit after providing the letter, then it's obvious you will be truthful on the application saying you've never been denied. A request for more information is not a denial -- it's just a temporary administrative hold.
Again, I'd withhold it to see if it's even needed. If you provide it, I can see the firearms section saying you need a newer one -- that a 10 year old letter is not "current."
Don't provide a reason for them to waste more of your time and money.
Of course, you can do as you wish. Not telling you to do it one way or the other. This is how I would do it.
If HMSA doesn't flag you, it doesn't matter what HPD has in your file. They will run a background check as usual. If they find nothing, there is nothing. Permit apporved.

You're worrying about answering a question that may never be asked. Cross that bridge when you come to it -- not before.