Ever watch Pawn Stars?
"My expert says it's worth $3500. I'll give you $1750 for it. Hey, I got overhead".
That is the experience you are going to have at your local gun store, at best. At worst, they will see it as just an old 1911 and offer you $300 because they are selling a lot of plastic wonder nines right now and they think it will sit on the shelf taking up space.
I am sorry to hear you are in a position where you are considering selling this piece, regardless of what the back story is. If it were mine, I would try to liquidate something else if possible and pass the piece along to someone in the family who will treat it as the family heirloom that it is. But if you are adamant about selling it for your own reasons, you want to maximize return on the item.
I see two options to get best dollar for it:
1) List on the auction site Gun Broker. Unless you are hard-up for cash immediately, take a little while and review many listings to get a feel for how best to list your gun here. I suspect you will get maximum value for it here so long as it is listed with very good pictures. The pictures you have are pretty decent, good enough for us to be able to make out important details, but for an auction listing the better the pics the better chance you have a maximizing value. Take the pistol outside in the late afternoon just as the sun is setting, put it on a table with a solid neutral color table cloth on it, and get a good point-n-shoot camera and take many snaps from many angles. You should have enough good light to not have to use a tripod. If the day is a little overcast, the pics will actually show detail a little better. Try to frame the shots (or crop them afterwards) to just show the gun on the table cloth. Make sure you take some pics with the camera set to "Macro" (usual symbol for Macro setting is a flower) and take good close-up shots of any and all scratches or dings or perceived flaws. Get good pics of the magazine as well, that may add value to the piece if it is correct. The downside to listing on Gun Broker is the potential for fraud and unscrupulous bidders.
2) List the pistol here once you get enough postings. You have a group of well-informed people here who CARE about these kinds of finds, and you are likely to find a person who is going to put it into their collection and treasure it instead of trying to flip it for a profit. If you want to know how much it should sell for, list it on Gun Broker with a $15k reserve and a $0.01 starting bid and see what the max bid is. Re-list it a few times to see what kind of bids you get, then list it here at somewhere around that amount. For a piece like this, I bet an interested buyer would even drive out to meet you Face-To-Face... I know I would if I could afford something like this.
If you can line up a Face-To-Face transaction, it should limit the risk of having to deal with a fraudulent buyer. Just make sure you have back-up on your side for any face-to-face meet like a couple of large male family members. However, to get top dollar you may have to ship it to an out-of-state buyer. If willing to ship, ask your local gun store what they charge to ship a pistol to another FFL (short for "Federal Firearm License holder", in other words another dealer). Should be around $50 or so, but it would be the legal way to transfer an item out to an out of state buyer. Ask the buyer to send a US Postal Money Order, which is the most secure form of non-cash payment. When you receive the money order, go to this page and follow instructions under "Accepting Money Orders" (https://www.usps.com/shop/money-orders.htm).
Once you are satisfied that the money order is good, then take the pistol to your local gun store and ask them to ship it for you. They should give you a tracking number, which you then pass along to the buyer so you both can track progress.
One other thing, if you are truly interested in selling it, consider investing in a letter from Colt for it. An expedited letter is $175, or a "regular" letter is $100 (same thing, just could take a lot longer to get the letter to you). The letter will provide very important provenance to the piece and will help to assure a buyer that what you have is the "real deal". Add a pic of this letter to your listing, whether here or on Gun Broker, put a sticky over personal info so you don't encourage someone to try and find you in real life, which could end poorly.
You have something that, if real, is going to pull in a fair amount of money. Don't leave money on the table by dealing with "Rick and Chumlee". Use this community to ask lots of questions, you will find the people here to be very helpful with a ton of knowledge.
Of course, you can reward the help you are getting here by choosing option 2
Best of luck!