If you marry, then you have a wedding. An elopement is still a wedding.
A wedding doesn't have to be large or elaborate, or cost a lot of money. It only costs as much as YOU choose to spend on it. Dinner and dancing isn't required, liquor doesn't need to be served, you don't have to have bridesmaids and groomsmen, she doesn't have to wear a white gown and a veil, etc. Your options aren't solely limited to "Huge expensive party" or "Private elopement" ... there is TONS of room in between these two extremes for you to plan for yourselves.
The only requirements are to get the marriage license from the state; to find a legally certified person to perform your ceremony (religious minister, judge, professional wedding officiant, friend ordained online, etc.); and if you're inviting guests then you need to seat and feed them (and that can be as basic as a grocery store sheet cake and a fruit platter). That is IT. Anything aside from those things is 100% optional.
Tell her that you'd really like to have your families and friends there when you marry, but you also understand her sadness at losing her mom and grandma and you also understand that she doesn't want to spend a lot of money or make a lot of decisions. Ask if there's a way you can compromise with her on having a wedding with guests, but still making it as low-key as possible.
If I were in your shoes, here are the two options I'd propose to her:
1. Cap the guest list at 50-100 people max. Find a local restaurant with a private banquet room that can put together an affordable lunch or brunch menu for you - just tell them what you want to spend and let them draft a menu and include nice table linens and centerpieces or candles in your package. Plan it for a month or two in the future. Get a marriage license. Send out simple invitations (you can order them online or buy them today at Target) or evites to your guests. Get a nice suit and a nice dress (not even white, unless she wants white) from a department store. No bridal party. Ask a florist or a grocery store flower department to put together a nice bouquet and boutonniere (you don't even have to pick the flowers ... just say "I can spend $x so do what you want and just make it look nice"), or skip the flowers entirely if you don't care about them. Hire an officiant to come and perform the ceremony at the restaurant, or ask a friend to get ordained online to do a short ceremony. Ask the restaurant to put on a background playlist of classical music or easy listening/Frank Sinatra-style stuff, and no dancing. If you want rings you can get them quickly from the stock options at a jewelry store, antique shop, department store, pawn shop, or Walmart, or order them online from Overstock.com or BlueNile.com. Boom, done, lovely wedding. My good friends threw this kind of wedding and it was absolutely beautiful, fairly stress-free, and didn't cost them an arm and a leg. If I had to do it over again, this is what I would do myself.
2. Hold a backyard cookout or a casual party at a VFW hall or firehouse hall. Or even something fun like a brewery, a bowling alley, an old movie theater, whatever - any place that has room to accommodate everyone, SEATED, and where you can provide some food. Tell your guests that the party is for July 4, or a family reunion, or "just because," or whatever excuse you want to use. Get a marriage license (which may require a witness, so if it's a secret wedding then make sure the witness you ask can keep a secret.) Arrange for barbecue food, soft drinks, and dessert - or if you do a quick mid-afternoon party you can get away with just serving a big sheet cake, a fruit platter, maybe a cheese platter, and soft drinks or punch. Make sure there are enough tables and chairs for every single person. Wear something casual but nice, like a sundress for her and a button-up shirt for you. Arrange to have someone certified to perform weddings in attendance. When everyone has arrived and is mingling, stand up and ask for their attention - thank them for coming and say, "We're so glad you're all here today, because we've decided to get married - right now!" Then ask them all to please take their seats, and have the certified officiant come forward and perform a short wedding ceremony. Then go on with the celebration as normal. That way you won't have people questioning your choices during the engagement and once they realize it's a wedding, it's over and done with. I have a friend who threw a semi-formal engagement party and then did a surprise wedding in the middle of it because they didn't want all the fuss of a big wedding.
And while you both need to agree with the plans, there's nothing written in stone about the woman being the one to spearhead them. It's perfectly OK for YOU to do the legwork and then say to her, "Do you like Option A or Option B best for our wedding?" or even "I'd like to book XYZ for our wedding, unless you have any objections."
But if she utterly refuses to marry you unless it's an elopement or a private wedding at a courthouse or whatever, then you have a decision to make ... is it more important to you to marry her no matter what, or more important to you to have your loved ones in attendance? You may have to make a very tough decision, and a sacrifice (either the wedding or her), if she refuses to compromise. (And, frankly, you may want to reconsider your entire relationship with someone who insists on their way or the highway.)