First off... here's a hug.
Your a amazing Dad, wanting to give and teach your kids values, as well teaching them about money and hard work. As well you have been through a horrible ordeal losing your wife, and the kids their mom. Congratulations on the promotion and raise. You are moving in the right direction. Being a single mom for several years, I understand the amount of pressure that you are under, with kids to take care of, the house, and work... how to balance it all... is over whelming. Please don't take this the wrong way, I do not mean any disrespect. When I was a single parent I belonged to a group like parents without partners which was truly a blessing to have that outlet, having someone to listen and learn, get ideas, and talk about things. Perhaps have you thought about maybe talking to someone at church or a grief group to help with the adult emotions that you have been through. I say this as our good friend lost his wife and close to the same situation as yourself. He just focused on his kids and never gave himself a moment to grieve, he was at work and had chest pains, they thought it was a heart attack at first then they realized it was an anxiety attack. Once he started talking to other people it gave him other adults to talk to, along with support and outlet for his emotions, and sounding board.
Possibly the family offering is letting each child take a friend so that they have someone to hang out with and will not argue with each other. That's what my parents did when we were kids. It's probably as simple as that. Could you be making more out of the offer, than what it really is and offer to take your kid to Disney?
For me I would let him go... As long as you are okay with them, and know that they will not let him run wild. Look at from his point, this is a chance to hangout with my friends, and more than likely not understand why you won't let him go. If another child goes and they talk about in front of your child how will he feel, left out more than likely. Which might cause problems at home and school.
The shame of things is not asking or accepting help, when you need it. Because you let pride stand in the way. If you knew someone was having a hard time and needed help would you offer and give it? It's the same thing.
Offering prayers and pixie dust.