Whatever you do, you need to make sure you have good evidence of what you have (photos, inventory, etc), and, if possible, an appraisal.
A safe is NOT required to be insured, but it's not a bad idea in general, and may? get you a discount with your insurance company if you can prove it. Just make sure it is bolted to the floor if it is less than 600 pounds empty. And big box store safes are generally worthless for theft prevention unless they have a theft rating.
I was burglarized about 10 years back, and my homeowners insurance would only cover face value. I didn't have much of a collection at that point, but I had *just* purchased the 2011 25th ASE set. So, I couldn't claim anything on it except that "several silver coins were stolen". Bleah.
Like for jewelry (think those giant engagement rings), you're going to need to get a rider from your home/renters insurance (or a separate policy) to cover the full value of your collection. That is NOT covered in your homeowners/renters policy, so you're going to have to pay more to insure your collection. The problem is, it's usually pretty costly for those riders or seperate policies. You have to do the math to figure out if it's worth paying the insurance. Or do you pay the bank and put your expensive stuff in their deposit boxes? Or do you buy your own safe (like I may have)? Or do you just risk it and hide your stuff under your mattress?
After collecting as long as I have....and knowing what my collection is worth... I've decided to self-insure with a rather large safe. I paid about $1,500 for it shortly after I was burglarized. It is more than adequate protection for what I have and also keeps my important documents protected.
So, there is no "right" answer. Depends on your collection and your wishes. Do your research. Evaluate what you have to help you make your decision.
And let us know what you end up doing. Backstory