The awkwardness of trying to celebrate a birthday with friends and define expenses you are willing and able to be responsible for is difficult. I have 8 kids now ages 12-31, so I found it helpful to "discuss it" with myself. What's bad about going to a birthday party where everything is not going to be "provided" or where it might be awkward for everything to be "provided" for every additional family member who might show up. Unexpected expenses are a little hard to deal with at the last minute. If I'm in the situation of your guests, and headed to a pool party where I KNOW that I will need to supervise my invited child and pay extra for my other children, it's easier to handle. What's rough is if I don't have enough money with me, or I have too many children with me. When you make out the invitations if you can mention the cost of the pool, and apologize for your inability to cover the total expense, that will alert the other moms and dads of the situation. I might suggest that you ask the pool if you can pay for a group rate of some sort so you can graciously include all the children who might need to be there. Some parents have local grandparents they can leave the other children with, but since you are a military wife yourself, I'm sure you know that it can be rough to pay a sitter at every turn. If the guest list is not too long perhaps you can invite a smaller number of whole families, and make it a family celebration. We did that a couple of times combining 2 sons whose birthdays and friends lists were close. The best party was the Pirate themed party with all the brothers and sisters at a park, friends helped me bring food, we provided some, the invitations were issued by phone so I could explain all the needs. For a gift suggestion I invited them to bring pennies to contribute to a "treasure chest" gift that my husband hid and made clues for a hunt. All the kids were on different teams for the hunt with an adult and teen helping. The money inside was divided between the two boys. This wasn't a little kids party, we all had fun, the siblings were all included, our friends helped us by bringing food and we all had a great time.
I think however you do it, avoiding the unexpected for your guests will make it pleasant and happy for everyone.