Good advice from Jaybay too and I was also thinking a 2 year old dog should not have to be going out during the night unless it is being fed or drinking a lot of water before bed time. I would have that checked out also. If you can't get a urine specimen, the Vet can do a draw with a needle (never bothered my dogs) to get a sample.
I do believe in crates and all my dogs have been crate trained. However, they were never crated for probably longer than 4 hrs, if need be. What I might do is try to baby gate them off in a kitchen area, if possible. Assuming the kitchen would have no carpet. You have to make sure it is dog safe with no cleaning products out, no trash out, no cords, food, etc. It might work if you left Dino out (assuming he can be left free roam) and put the little one behind a baby gate. It may just be trial and error to see what works best for them and you.
I'm more concerned that your dog is having accidents at night. Since it appears this only happens at night it doesn't sound like a housetraining issue. Once a dog becomes a young adult, he is perfectly capable of holding his waste overnight as long as he's let out to empty his tanks before bedtime. Before you buy a crate, please take your dog and a fresh urine sample to your vet. He may have a urinary tract infection and simply can't hold it through the night. If that's the case, it's easily treated with antibiotics.
Crate training is a good thing for any dog. The idea is to make the crate the coolest, most attractive and relaxing place for your dog to chill out. It works with a dog's natural instincts to curl up in a den to rest and go off-duty from guarding his territory. Make it comfortable with a thick pad or blanket and toss in a soft piece of clothing that you've worn. Having your scent in there with him is a comfort.
Start with a game of "fetch the treat" by tossing a treat in the crate and praising him when he goes in to retrieve it. Do that for a day or two.
Next, ask him to stay in the crate while he eats the treat. Don't shut the door, but use your body to block him from exiting. Ask him to lie down, praise and treat again. Once he's calm, invite him to exit. Once he's comfortable doing that, close the door for a few minutes and treat through the door. Only invite him out when he's calm and relaxed. Restrict your talking to only giving commands or praise. Chatter only exites the dog and confuses the issue. You want the experience to be a good one and matter-of-fact. If you act like it's a Big Deal, your dog will think it's a Big Deal and something to be avoided.
Another way to make the crate the place from which all blessings flow is to feed your dog in the crate. You can do that either with the door open or closed depending on how comfortable he is with the crate. Keep up with the treat exercises and gradually extend the amount of time your dog is crated. Avoid exited chatter when you release him. You don't want your dog to anticipate excitement and throw himself around in the crate and risk an injury.
During the day, keep the crate in an easily accessible spot that he can retreat to periodically. One of my dogs is scared of fireworks and thunder. She takes herself off to her crate to calm herself down. The world is much smaller and easier to deal with in a comfortable hidey hole. Hope that helps!